

A336657


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


1



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
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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 dk 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. 403413.


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 A094529
Adjacent sequences: A336654 A336655 A336656 * A336658 A336659 A336660


KEYWORD

nonn


AUTHOR

Amiram Eldar, Jul 28 2020


STATUS

approved



