© 1996-2008 Robert P. Munafo. Email the author
This work is licensed under a
Creative Commons Attribution 2.5 License.
| Some Integer Sequences |
This page gives terms in some of the integer sequences mentioned on my numbers and large numbers pages.
If this sort of thing is of interest, you should also take a look at my comprehensive list of sequences generated by "classical" formulas like "AN = 2N+1" or "A0=1; AN+1 = AN+2".
A052154: Coefficents of Lemniscates for Mandelbrot Set
2, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 5, 0, 0, 0, 1, 1, 2, 5, ...
This sequence has its own page here.
Links to the mu-ency title page and to my A052154 page.
A019473: Still-Lifes with N cells in Conway's game of Life
2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, ...
This sequence has its own page here.
I am listed as author. I believe I sent this to NJAS in the initial email with sequences to add after his 1995 book.
A092188: Smallest positive integer M such that 2^3^4^5^...^N ≡ M mod N
2, 2, 4, 2, 2, 1, 8, 8, 2, 2, 8, 5, 8, 2, 16, 2, 8, 18, 12, ...
This sequence, Sloane's A092188, has its own page here.
Links to my page on the topic, and mentions that I extended the sequence in 200404.
A080611: Smallest base with divisibiliy tests for first N primes
2, 2, 4, 6, 21, 155, 441, 2925, 10165, 342056, 2781505, ...
My numbers page is in the links.
A000027: Natural numbers
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ...
My numbers page is in the links.
A006874: Period-N Continental mu-atoms in Mandelbrot Set
2, 3, 4, 6, 6, 9, 10, 12, 10, 22, 12, 18, 24, 27, 16, 38, ...
This sequence is discussed here and here.
This sequence links to the mu-ency title page.
A006875: Period-N Non-seed mu-atoms in Mandelbrot Set
2, 3, 4, 7, 6, 12, 12, 23, 10, 51, 12, 75, 50, 144, 16, ...
This sequence is discussed here and here.
This sequence links to the mu-ency title page.
A000040: Prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, ...
A000041: Partitions
2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, ...
Sloane's A000041, another combinatoric sequence like the Bell numbers; "ways to put N indistinguishable balls into one or more indistinguishable urns". More terms here.
A000045: Fibonacci numbers
2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
More terms here.
A006888: An = An-1 + An-2 An-3
2, 3, 5, 11, 26, 81, 367, 2473, 32200, 939791, 80570391, 30341840591, 75749670168872, ...
I discovered this with the Casio 8500fx calculator, and submitted it to NJAS along with several other sequences.
A006877: Successive values of X1 for Collatz 3X+1 iteration that set a new record for Np
2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, ...
This sequence is discussed here.
I am listed as author; I calculated this sequence on an Apple II shortly after the publication of Goedel, Escher, Bach and submitted it to Sloane after his 1995 book came out.
A069354: Lowest base with easy test for divibility by N primes
2, 3, 6, 15, 66, 210, 715, 7315, 38571, 254541, 728365, ...
Lists me as author and links to the numbers page.
A006884: Successive values of X1 for Collatz 3X+1 iteration that set a record for Xmax
2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, ...
This sequence is discussed here.
I am listed as author; I calculated this sequence on an Apple II shortly after the publication of Goedel, Escher, Bach and submitted it to Sloane after his 1995 book came out.
A006882: Double Factorials
2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, ...
Combined odd and even double factorials. I am listed as the author.
A014221: 2^^n
2, 4, 16, 65536, ...
My page on A094358 is listed as a link.
A000110: Bell numbers
2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, ...
Sloane's A000110, this sequence shows up in many combinatoric applications; the simplest description is "ways to put N distinguishable balls into one or more indistinguishable urns". More terms here
A100140: Largest Denominator of Greedy Egyptian Fraction Sum for M/N
2, 6, 4, 20, 3, 231, 24, 45, 20, 4070, 12, 2145, 231, 120, ...
This sequence, Sloane's a100140, has its own page here.
Links to my A100140 page and lists me as author (I submitted this in 200411, during the initial Filene's investigation of greedy Egyptian fractions)
A006883: 1/N has repeating decimal of N-1 digits
2, 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, ...
I am listed as author; this is one of the sequences in my original list of things to add to Sloane's 1995 book.
A054670: Laurent series for mapping onto boundary of Mandelbrot set (Denominators)
2, 8, 4, 128, 1, 1024, 16, 32768, 1, 262144, 32, 4194304, ...
Lists me as author, and links to my mu-ency page on the Laurent series for the Maple code.
A006885: Successive record values of Xmax for Collatz 3X+1 iteration
2, 16, 52, 160, 9232, 13120, 39364, 41524, 250504, 1276936, ...
This sequence is discussed here.
I am listed as author; I calculated this sequence on an Apple II shortly after the publication of Goedel, Escher, Bach and submitted it to Sloane after his 1995 book came out.
A000203: Sum of all divisors of N
3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, ...
Sloane's A000203, the sum of all divisors of N. More terms here
A080307: Multiples of Fermat numbers
3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 24, 25, 27, 30, 33, ...
Links to my notes on Fermat numbers.
A094358: 2④N ≡ 1 mod N
3, 5, 15, 17, 51, 85, 255, 257, 641, 771, 1285, 1923, 3205, 3855, 4369, 9615, 10897, 13107, 21845, 32691, 54485, 65535, 65537, 114689, 163455, 164737, 196611, ...
This sequence, Sloane's A094358, has its own page here.
Links to my page on the topic, lists me as author and mentions the conjecture linking the sequence to A023394.
A023394: Prime Factors of Fermat Numbers
3, 5, 17, 257, 641, 65537, 114689, 274177, 319489, 974849, ...
This sequence has its own page here.
A050922: Fermat Numbers, factorized
3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, ...
Like A023394, this sequence links to my fermat numbers page.
A000215: Fermat numbers
3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ...
This sequence has its own page here.
A000740: Period-N mu-atoms in Mandelbrot Set
3, 6, 15, 27, 63, 120, 252, 495, 1023, 2010, 4095, 8127, ...
This sequence is discussed here and here.
I submitted this to NJAS in early 2000, I am listed as having "proved" the connection to the Mandelbrot set (although the majority of the work had already been done by others who explored and described the complex dynamics of the iterated quadratic functions that define the Julia sets)
A006882: Double Factorials
3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, ...
A006876: Period-N mu-molecules in Mandelbrot Set
3, 11, 20, 57, 108, 240, 472, 1013, 1959, 4083, 8052, ...
This sequence is discussed here and here.
This sequence links to the mu-ency title page.
A006890: Feigenbaum constant
4, 6, 6, 9, 2, 0, 1, 6, 0, 9, 1, 0, 2, 9, 9, 0, 6, 7, 1, 8, ...
My mu-ency page on the Feigenbaum constant is listed as a link.
A098403: Area of Mandelbrot set
5, 0, 6, 5, 9, 1, ...
My mu-ency page on the Area of the Mandelbrot Set is listed as a link.
A078335: e^X exceeds X!
5, 2, 9, 0, 3, 1, 6, 0, 9, 3, 1, 1, 9, 7, 7, 0, 7, 1, 0, 7, ...
This sequence links to my numbers page.
A078333: Sqrt(2)^sqrt(2)
6, 3, 2, 5, 2, 6, 9, 1, 9, 4, 3, 8, 1, 5, 2, 8, 4, 4, 7, 7, ...
This sequence links to my numbers page.
A006881: Product of two distinct primes
6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, ...
I am listed as a co-author, I probably called attention to it in connection with A055233.
A019296: Values of N for which eπ sqrt(N) is nearly an integer
6, 17, 18, 22, 25, 37, 43, 58, 59, 67, 74, 103, 148, 149, ...
This sequence, Sloane's a019296, and several related sequences have their own page here.
A006878: Successive record values of Np for Collatz 3X+1 iteration
7, 8, 16, 19, 20, 23, 111, 112, 115, 118, 121, 124, 127, ...
This sequence is discussed here.
I am listed as author; I calculated this sequence on an Apple II shortly after the publication of Goedel, Escher, Bach and submitted it to Sloane after his 1995 book came out.
A094534: Centered Hexamorphic Numbers
7, 17, 51, 67, 167, 251, 417, 501, 667, 751, 917, 1251, ...
This sequence, Sloane's a094534, has its own page here.
Lists me as author, and links to my A094534 page.
A006887: Kaprekar numbers for cubes, omitting 10x
8, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, ...
This sequence is discussed here.
I am listed as author; I submitted this the same time as the normal (square) Kaprekar numbers.
A006886: Kaprekar numbers for squares, allowing leading 0
9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, ...
This sequence is discussed here.
I am listed as author. I found the Kaprekar numbers in Wells and sent them to Sloane with many other suggestions after his 1995 book.
A053816: Kaprekar numbers for squares
9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, ...
This sequence is discussed here.
As with A006886, I am listed as author.
A055233: N = P + ... + Q, P and Q are smallest and largest prime factors of N
10, 39, 155, 371, 2935561623745, ...
I was the first to give an extension to the "39-property", the 454539357304421 term, on March 7, 2000. See this article in sci.math.
Sloane also links to a primepuzzles.net article. If you look at the dates, it appears that Jud McCranie wrote on January 7th, March 7th and July 7th 2000. However if you look here or on any of a number of other Prime Puzzles articles, you can see that the dates are given as Day/Month/Year — so in article 98, Jud McCranie actually write on the 1st, 3rd and 7th of July.
I consider the problem of N = P Q = P + ... + Q to be more interesting, but Sloane does not yet have an entry for that. If he did, 454539357304421 would be the next term after 10, 39, 155 and 371.
A054670: Laurent series for mapping onto boundary of Mandelbrot set (Numerators)
15, 0, -47, -1, 987, 0, -3673, 1, -61029, 0, -689455, -21, ...
Lists me as author, and links to my mu-ency page on the Laurent series for the Maple code.
A100933: Numbers having more than one representation as the product of consecutive integers > 1
120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, ...
This sequence has its own page here.
Gives my extension from 20070813; still waiting for my more recent extension to be added.
A000203: Sum of all divisors of N
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140, 96, 168, 80, 186, 121, 126, 84, 224, 108, 132, 120, 180, 90, 234, 112, 168, 128, 144, 120, 252, 98, 171, 156, 217, 102, 216, 104, 210, 192, 162, 108, 280, 110, 216, 152, 248, 114, 240, 144, 210, 182, 180, 144, 360, 133, 186, 168, 224, 156, 312, 128, 255, 176, 252, 132, 336, 160, 204, 240, 270, 138, 288, 140, 336, 192, 216, 168, 403, 180, 222, 228, 266, 150, 372, 152, 300, 234, 288, 192, 392, 158, 240, 216, 378, 192, 363, 164, 294, 288, 252, 168, 480, 183, 324, 260, 308, 174, 360, 248, 372, 240, 270, 180, 546, 182, 336, 248, 360, 228, 384, 216, 336, 320, 360, 192, 508, 194, 294, 336, 399, 198, 468, 200, 465, 272, 306, 240, 504, 252, 312, 312, 434, 240, 576, 212, 378, 288, 324, 264, 600, 256, 330, 296, 504, 252, 456, 224, 504, 403, 342, 228, 560, 230, 432, 384, 450, 234, 546, 288, 420, 320, 432, 240, 744, 242, 399, 364, 434, 342, 504, 280, 480, 336, 468, 252, 728, 288, 384, 432, 511, 258, 528, 304, 588, 390, 396, 264, 720, 324, 480, 360, 476, 270, 720, 272, 558, 448, 414, 372, 672, 278, 420, 416, 720, 282, 576, 284, 504, 480, 504, 336, 819, 307, 540, 392, 518, 294, 684, 360, 570, 480, 450, 336, 868, 352, 456, 408, 620, 372, 702, 308, 672, 416, 576, 312, 840, 314, 474, 624, 560, 318, 648, 360, 762, 432, 576, 360, 847, 434, 492, 440, 630, 384, 864, 332, 588, 494, 504, 408, 992, 338, 549, 456, 756, 384, 780, 400, 660, 576, 522, 348, 840, 350, 744, 560, 756, 354, 720, 432, 630, 576, 540, 360, 1170, 381, 546, 532, 784, 444, 744, 368, 744, 546, 684, 432, 896, 374, 648, 624, 720, 420, 960, 380, 840, 512, 576, 384, 1020, 576, 582, 572, 686, 390, 1008, 432, 855, 528, 594, 480, 1092, 398, 600, 640, 961, 402, 816, 448, 714, 726, 720, 456, 1080, 410, 756, 552, 728, 480, 936, 504, 882, 560, 720, 420, 1344, 422, 636, 624, 810, 558, 864, 496, 756, 672, 792, 432, 1240, 434, 768, 720, 770, 480, 888, 440, 1080, 741, 756, 444, 1064, 540, 672, 600, 1016, 450, 1209, 504, 798, 608, 684, 672, 1200, 458, 690, 720, 1008, 462, 1152, 464, 930, 768, 702, 468, 1274, 544, 864, 632, 900, 528, 960, 620, 1008, 702, 720, 480, 1512, 532, 726, 768, 931, 588, 1092, 488, 930, 656, 1026, 492, 1176, 540, 840, 936, 992, 576, 1008, 500, 1092, 672, 756, 504, 1560, 612, 864, 732, 896, 510, 1296, 592, 1023, 800, 774, 624, 1232, 576, 912, 696, 1260, 522, 1170, 524, 924, 992, 792, 576, 1488, 553, 972, 780, 1120, 588, 1080, 648, 1020, 720, 810, 684, 1680, 542, 816, 728, 1134, 660, 1344, 548, 966, 806, 1116, 600, 1440, 640, 834, 912, 980, 558, 1248, 616, 1488, 864, 846, 564, 1344, 684, 852, 968, 1080, 570, 1440, 572, 1176, 768, 1008, 744, 1651, 578, 921, 776, 1260, 672, 1176, 648, 1110, 1092, 882, 588, 1596, 640, 1080, 792, 1178, 594, 1440, 864, 1050, 800, 1008, 600, 1860, 602, 1056, 884, 1064, 798, 1224, 608, 1260, 960, 1116, 672, 1638, 614, 924, 1008, 1440, 618, 1248, 620, 1344, 960, 936, 720, 1736, 781, 942, 960, 1106, 684, 1872, 632, 1200, 848, 954, 768, 1512, 798, 1080, 936, 1530, 642, 1296, 644, 1344, 1056, 1080, 648, 1815, 720, 1302, 1024, 1148, 654, 1320, 792, 1302, 962, 1152, 660, 2016, 662, 996, 1008, 1260, 960, 1482, 720, 1176, 896, 1224, 744, 2016, 674, 1014, 1240, 1281, 678, 1368, 784, 1620, 912, 1152, 684, 1820, 828, 1200, 920, 1364, 756, 1728, 692, 1218, 1248, 1044, 840, 1800, 756, 1050, 936, 1736, 702, 1680, 760, 1524, 1152, 1062, 816, 1680, 710, 1296, 1040, 1350, 768, 1728, 1008, 1260, 960, 1080, 720, 2418, 832, 1143, 968, 1274, 930, 1596, 728, 1680, 1093, 1332, 792, 1736, 734, 1104, 1368, 1512, 816, 1638, 740, 1596, 1120, 1296, 744, 1920, 900, 1122, 1092, 1512, 864, 1872, 752, 1488, 1008, 1260, 912, 2240, 758, 1140, 1152, 1800, 762, 1536, 880, 1344, 1404, 1152, 840, 2044, 770, 1728, 1032, 1358, 774, 1716, 992, 1470, 1216, 1170, 840, 2352, 864, 1296, 1200, 1767, 948, 1584, 788, 1386, 1056, 1440, 912, 2340, 868, 1194, 1296, 1400, 798, 1920, 864, 1953, 1170, 1206, 888, 1904, 1152, 1344, 1080, 1530, 810, 2178, 812, 1680, 1088, 1368, 984, 2232, 880, 1230, 1456, 1764, 822, 1656, 824, 1560, 1488, 1440, 828, 2184, 830, 1512, 1112, 1778, 1026, 1680, 1008, 1680, 1280, 1260, 840, 2880, 871, 1266, 1128, 1484, 1098, 1872, 1064, 1674, 1136, 1674, 912, 2016, 854, 1488, 1560, 1620, 858, 2016, 860, 1848, 1344, 1296, 864, 2520, 1044, 1302, 1228, 1792, 960, 2160, 952, 1650, 1274, 1440, 1248, 2072, 878, 1320, 1176, 2232, 882, 2223, 884, 1764, 1440, 1332, 888, 2280, 1024, 1620, 1452, 1568, 960, 1800, 1080, 2040, 1344, 1350, 960, 2821, 972, 1512, 1408, 1710, 1092, 1824, 908, 1596, 1326, 2016, 912, 2480, 1008, 1374, 1488, 1610, 1056, 2160, 920, 2160, 1232, 1386, 1008, 2688, 1178, 1392, 1352, 1890, 930, 2304, 1140, 1638, 1248, 1404, 1296, 2730, 938, 1632, 1256, 2016, 942, 1896, 1008, 1860, 1920, 1584, 948, 2240, 1036, 1860, 1272, 2160, 954, 2106, 1152, 1680, 1440, 1440, 1104, 3048, 993, 1596, 1404, 1694, 1164, 2304, 968, 1995, 1440, 1764, 972, 2548, 1120, 1464, 1736, 1922, 978, 1968, 1080, 2394, 1430, 1476, 984, 2520, 1188, 1620, 1536, 1960, 1056, 2808, 992, 2016, 1328, 1728, 1200, 2352, 998, 1500, 1520, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, 474869816156751, 4506715738447323, 44152005855084346, 445958869294805289, 4638590332229999353, ...
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, 1779979416004714189, 2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738, 19740274219868223167, 31940434634990099905, 51680708854858323072, 83621143489848422977, 135301852344706746049, 218922995834555169026, 354224848179261915075, 573147844013817084101, 927372692193078999176, 1500520536206896083277, 2427893228399975082453, 3928413764606871165730, 6356306993006846248183, 10284720757613717413913, 16641027750620563662096, 26925748508234281076009, 43566776258854844738105, 70492524767089125814114, 114059301025943970552219, 184551825793033096366333, 298611126818977066918552, 483162952612010163284885, 781774079430987230203437, 1264937032042997393488322, 2046711111473984623691759, 3311648143516982017180081, 5358359254990966640871840, 8670007398507948658051921, 14028366653498915298923761, 22698374052006863956975682, 36726740705505779255899443, 59425114757512643212875125, 96151855463018422468774568, 155576970220531065681649693, 251728825683549488150424261, 407305795904080553832073954, 659034621587630041982498215, 1066340417491710595814572169, 1725375039079340637797070384, 2791715456571051233611642553, 4517090495650391871408712937, 7308805952221443105020355490, 11825896447871834976429068427, 19134702400093278081449423917, 30960598847965113057878492344, 50095301248058391139327916261, 81055900096023504197206408605, 131151201344081895336534324866, 212207101440105399533740733471, 343358302784187294870275058337, 555565404224292694404015791808, 898923707008479989274290850145, 1454489111232772683678306641953, 2353412818241252672952597492098, 3807901929474025356630904134051, 6161314747715278029583501626149, 9969216677189303386214405760200, 16130531424904581415797907386349, 26099748102093884802012313146549, 42230279526998466217810220532898, 68330027629092351019822533679447, 110560307156090817237632754212345, 178890334785183168257455287891792, 289450641941273985495088042104137, 468340976726457153752543329995929, 757791618667731139247631372100066, 1226132595394188293000174702095995, 1983924214061919432247806074196061, 3210056809456107725247980776292056, 5193981023518027157495786850488117, 8404037832974134882743767626780173, 13598018856492162040239554477268290, 22002056689466296922983322104048463, 35600075545958458963222876581316753, 57602132235424755886206198685365216, 93202207781383214849429075266681969, 150804340016807970735635273952047185, 244006547798191185585064349218729154, 394810887814999156320699623170776339, 638817435613190341905763972389505493, 1033628323428189498226463595560281832, 1672445759041379840132227567949787325, 2706074082469569338358691163510069157, 4378519841510949178490918731459856482, 7084593923980518516849609894969925639, 11463113765491467695340528626429782121, 18547707689471986212190138521399707760, 30010821454963453907530667147829489881, 48558529144435440119720805669229197641, 78569350599398894027251472817058687522, 127127879743834334146972278486287885163, ...
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