A336657 Numbers k such that 2^k - 1 is divisible by the sum of the distinct primes dividing k (A008472).

12, 24, 36, 48, 52, 72, 96, 104, 108, 144, 192, 208, 216, 288, 324, 330, 345, 384, 385, 416, 432, 462, 576, 648, 660, 664, 665, 676, 690, 768, 832, 840, 864, 924, 972, 990, 1035, 1152, 1190, 1296, 1302, 1320, 1328, 1330, 1352, 1380, 1386, 1428, 1430, 1530, 1536

Offset

1,1

Comments

Since 2^p == 2 (mod p) for all primes p, all the terms of this sequence are composites. Similar considerations show that there are no semiprimes in this sequence.

The odd terms are relatively rare: 345, 385, 665, 1035, 1725, 1925, ...

If k is a term and d|k then d*k is also a term. In particular, all the numbers of the form 2^i * 3^j, with i > 1 and j > 0, are terms.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

William D. Banks and Florian Luca, Sums of prime divisors and Mersenne numbers, Houston J. Math., Vol. 33, No. 2 (2007), pp. 403-413.

Formula

The number of terms not exceeding x is x^(1 - c_1 * log(log(log(x)))/log(log(x))) <= N(x) <= c_2 * x * log(log(x))/log(x) for all sufficiently large values of x, where c_1 and c_2 are positive constants (Banks and Luca, 2007).

Example

12 = 2^2 * 3 is a term since 2^12 - 1 = 4095 is divisible by 2 + 3 = 5.

Mathematica

b[n_] := Total[FactorInteger[n][[;; , 1]]]; Select[Range[2, 1500], PowerMod[2, #, b[#]] == 1 &]

Crossrefs

Cf. A000225, A008472, A156787.

Sequence in context: A355455 A059691 A097060 * A066085 A340511 A382932

Adjacent sequences: A336654 A336655 A336656 * A336658 A336659 A336660

Keyword

nonn

Author

Amiram Eldar, Jul 28 2020

Status

approved