Sequence A007898: Related to Fibonacci Partitions and Products of Reduced Fractions

This sequence, Sloane's A7898, is closely related to the sequence A7898, both related to the Fibonacci partitions A0119 and found by Felix Weinstein who sent me a preprint of his paper ¹ in 1994.

A007898: 1, 1, 2, 3, 3, 4, 4, 7, 7, 6, 6, 12, 7, 8, 12, 16, 9, 15, 10, 18, 16, 12, 12, 32, 17, 14, 22, 24, 15, 30, 16, 34, 24, 18, 24, 48, 19, 20, 28, 48, 21, 40, 22, 36, 45, 24, 24, 78, 32, 37, 36, 42, 27, 54, 36, 64, 40, 30, 30, 96, 31, 32, 60, 78, 42, 60, 34, 54, ...

According to the OEIS page (A007898), the sequence can be generated by the following Pari/GP program (by Michael Somos, to which I added a minor syntax correction and the last line that actually prints out terms of the sequence):

/* To run this from the command line: echo '' | /path/to/pari-gp -q a7898.gp */ a(n) = { my(A, v, w, m, p, e); if( n<1, 0, v = vector(n, k, k==1); for(k=2, n, m = #digits(n, k) - 1; A = factor(k); A = prod( j=1, matsize(A)[1], if( p = A[j,1], e = A[j,2]; if( p==2, if( e<3, e, 2^(e-2) + 2), 1 + p^(e-1) * (p-1) / 2 ) ) ); A = (1 - x)^ -A + x * O(x^m); w = vector(n); for(i=0, m, w[k^i] = polcoeff(A, i) ); v = dirmul(v, w) ); v[n] ) }; for(i=1, 100, print1(a(i), ", ");); print("...")

My Variant Definition

I have discovered a closely-releated sequence, that agrees with this one until A(45), for which the currect value is 47 but I get 45. My definition is suggested by the concept of "*-equivalence" that is defined in the Weinstein paper ¹ as follows (paraphrased to be more obvious}:

The fraction-products a₁/b₁ × a₂/b₂ × ... × a_m/b_m and c₁/d₁ × c₂/d₂ × ... × c_m/d_m are referred to as:

"t-equivalent" if the set {a₁/b₁, a₂/b₂, ... a_m/b_m} is equivalent to the set {c₁/d₁, c₂/d₂, ... c_m/d_m}, i.e. the two are the same if reordered.

"*-equivalent" if the two fraction-products are t-equivalent and if [for all i] b_i=d_i then either a_i=c_i or a_i×c_i is 1 more than a multiple of d_i.

T-equivalence simply means that the two are the same if reordered. This is used to define A7896 as distinct from A6874, in which reordering is not allowed: for A7896 any reordering is considered to be the same fraction-product, but in A6874 all orderings are counted separately.

My definition of a sequence that is similar to A7898 considers two fractions a/d and c/d to be "equivalent" if a×b is 1 more than a multiple of d. This causes fewer fraction-products to be accepted for counting purposes whenever there n has a prime factor of 5 or more.

Here are the first 25 terms, with the fraction-products for each (compare to the A7896 terms):

A(1) = 1: (null product)
A(2) = 1: 1/2
A(3) = 2: 1/3; 2/3
A(4) = 3: 1/2×1/2; 1/4; 3/4
A(5) = 3: 1/5; 2/5; 4/5
A(6) = 4: 1/2×1/3; 1/2×2/3; 1/6; 5/6
A(7) = 4: 1/7; 2/7; 3/7; 6/7
A(8) = 7: 1/2×1/2×1/2; 1/2×1/4; 1/2×3/4; 1/8; 3/8; 5/8; 7/8
A(9) = 7: 1/3×1/3; 1/3×2/3; 2/3×2/3; 1/9; 2/9; 4/9; 8/9
A(10) = 6: 1/2×1/5; 1/2×2/5; 1/2×4/5; 1/10; 3/10; 9/10
A(11) = 6: 1/11; 2/11; 3/11; 5/11; 7/11; 10/11
A(12) = 12: 1/2×1/2×1/3; 1/2×1/2×2/3; 1/2×1/6; 1/2×5/6; 1/3×1/4; 1/3×3/4; 2/3×1/4; 2/3×3/4; 1/12; 5/12; 7/12; 11/12
A(13) = 7: 1/13; 2/13; 3/13; 4/13; 5/13; 6/13; 12/13
A(14) = 8: 1/2×1/7; 1/2×2/7; 1/2×3/7; 1/2×6/7; 1/14; 3/14; 9/14; 13/14
A(15) = 12: 1/3×1/5; 1/3×2/5; 1/3×4/5; 2/3×1/5; 2/3×2/5; 2/3×4/5; 1/15; 2/15; 4/15; 7/15; 11/15; 14/15
A(16) = 16: 1/2×1/2×1/2×1/2; 1/2×1/2×1/4; 1/2×1/2×3/4; 1/2×1/8; 1/2×3/8; 1/2×5/8; 1/2×7/8; 1/4×1/4; 1/4×3/4; 3/4×3/4; 1/16; 3/16; 5/16; 7/16; 9/16; 15/16
A(17) = 9: 1/17; 2/17; 3/17; 4/17; 5/17; 8/17; 10/17; 11/17; 16/17
A(18) = 15: 1/2×1/3×1/3; 1/2×1/3×2/3; 1/2×2/3×2/3; 1/2×1/9; 1/2×2/9; 1/2×4/9; 1/2×8/9; 1/3×1/6; 1/3×5/6; 2/3×1/6; 2/3×5/6; 1/18; 5/18; 7/18; 17/18
A(19) = 10: 1/19; 2/19; 3/19; 4/19; 6/19; 7/19; 8/19; 9/19; 14/19; 18/19
A(20) = 18: 1/2×1/2×1/5; 1/2×1/2×2/5; 1/2×1/2×4/5; 1/2×1/10; 1/2×3/10; 1/2×9/10; 1/4×1/5; 1/4×2/5; 1/4×4/5; 3/4×1/5; 3/4×2/5; 3/4×4/5; 1/20; 3/20; 9/20; 11/20; 13/20; 19/20
A(21) = 16: 1/3×1/7; 1/3×2/7; 1/3×3/7; 1/3×6/7; 2/3×1/7; 2/3×2/7; 2/3×3/7; 2/3×6/7; 1/21; 2/21; 4/21; 5/21; 8/21; 10/21; 13/21; 20/21
A(22) = 12: 1/2×1/11; 1/2×2/11; 1/2×3/11; 1/2×5/11; 1/2×7/11; 1/2×10/11; 1/22; 3/22; 5/22; 7/22; 13/22; 21/22
A(23) = 12: 1/23; 2/23; 3/23; 4/23; 5/23; 7/23; 9/23; 11/23; 13/23; 15/23; 17/23; 22/23
A(24) = 32: 1/2×1/2×1/2×1/3; 1/2×1/2×1/2×2/3; 1/2×1/2×1/6; 1/2×1/2×5/6; 1/2×1/3×1/4; 1/2×1/3×3/4; 1/2×2/3×1/4; 1/2×2/3×3/4; 1/2×1/12; 1/2×5/12; 1/2×7/12; 1/2×11/12; 1/3×1/8; 1/3×3/8; 1/3×5/8; 1/3×7/8; 2/3×1/8; 2/3×3/8; 2/3×5/8; 2/3×7/8; 1/4×1/6; 1/4×5/6; 3/4×1/6; 3/4×5/6; 1/24; 5/24; 7/24; 11/24; 13/24; 17/24; 19/24; 23/24
A(25) = 17: 1/5×1/5; 1/5×2/5; 1/5×4/5; 2/5×2/5; 2/5×4/5; 4/5×4/5; 1/25; 2/25; 3/25; 4/25; 6/25; 7/25; 8/25; 9/25; 11/25; 12/25; 24/25
...

1 : Felix V. Weinstein, "[Notes on Fibonacci partitions", 1994. Submitted to arXiv at math/0307150 in 2003; revised and expanded through 2018 or later.

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2024 Mar 19.

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