Sequence A094358, 2^^N ≡ 1 mod N

This sequence consists of all numbers N such that 2↑↑N ≡ 1 mod N.

2↑↑N, also written 2^④N, is a tower of 2's N high, a very quickly-growing function given by Sloane's A014221:

2↑↑0 = 1,
2↑↑1 = 2,
2↑↑2 = 2² = 4,
2↑↑3 = 2^2² = 16,
2↑↑4 = 2^{2^2²} = 65536,
2↑↑5 = 2^{2^{2^2²}} ≅ 2.00353×10¹⁹⁷²⁸,
2↑↑6 = 2^{2^{2^{2^2²}}} ≅ 10^{6.03122×10¹⁹⁷²⁷},
and so on.

N is in the sequence if 2↑↑N ≡ 1 mod N. For example, 3 is in the sequence because 2↑↑3 = 16, and 16 ≡ 1 mod 3.

Using the techniques described here it is easy to calculate the values of N such that 2↑↑N ≡ 1 mod N.

The sequence, A094358, starts: 1, 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, 274177, 319489, 327685, 344067, 494211, 573445, 822531, 823685, 958467, 974849, 983055, 1114129, 1370885, 1597445, 1720335, 1949713, 2424833, 2471055, 2800529, 2924547, 3342387, 4112655, 4661009, 4792335, 4874245, 5431313, 5570645, 5849139, 6700417, 7274499, 8401587, 9748565, 12124165, ...

As far as I've seen so far, each term is a squarefree product of terms in A023394, the prime factors of the Fermat numbers. The factorizations are:

3 (prime)

5 (prime)

15 = 3 × 5

17 (prime)

51 = 3 × 17

85 = 5 × 17

255 = 3 × 5 × 17

257 (prime)

641 (prime)

771 = 3 × 257

1285 = 5 × 257

1923 = 3 × 641

3205 = 5 × 641

3855 = 3 × 5 × 257

4369 = 17 × 257

9615 = 3 × 5 × 641

10897 = 17 × 641

13107 = 3 × 17 × 257

21845 = 5 × 17 × 257

32691 = 3 × 17 × 641

54485 = 5 × 17 × 641

65535 = 3 × 5 × 17 × 257

65537 (prime)

114689 (prime)

163455 = 3 × 5 × 17 × 641

164737 = 257 × 641

196611 = 3 × 65537

274177 (prime)

319489 (prime)

327685 = 5 × 65537

344067 = 3 × 114689

494211 = 3 × 257 × 641

573445 = 5 × 114689

822531 = 3 × 274177

823685 = 5 × 257 × 641

958467 = 3 × 319489

974849 (prime)

983055 = 3 × 5 × 65537

1114129 = 17 × 65537

1370885 = 5 × 274177

1597445 = 5 × 319489

1720335 = 3 × 5 × 114689

1949713 = 17 × 114689

2424833 (prime)

2471055 = 3 × 5 × 257 × 641

2800529 = 17 × 257 × 641

2924547 = 3 × 974849

3342387 = 3 × 17 × 65537

4112655 = 3 × 5 × 274177

4661009 = 17 × 274177

4792335 = 3 × 5 × 319489

4874245 = 5 × 974849

5431313 = 17 × 319489

5570645 = 5 × 17 × 65537

5849139 = 3 × 17 × 114689

6700417 (prime)

7274499 = 3 × 2424833

8401587 = 3 × 17 × 257 × 641

9748565 = 5 × 17 × 114689

12124165 = 5 × 2424833

(...)

At first I thought that the sequence was related to A058910 (because of its definition, being a large power of 2 mod N for some large N). The sequence is also similar to A001317 and A004729, until we get to 641. The appearance of 641 is what tipped me off to the possible link with the Fermat factors.

Some other sequences I have investigated are discussed here.

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