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 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 (list; graph; refs; listen; history; text; internal format)
 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 A094529 Adjacent sequences:  A336654 A336655 A336656 * A336658 A336659 A336660 KEYWORD nonn AUTHOR Amiram Eldar, Jul 28 2020 STATUS approved

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Last modified August 8 09:40 EDT 2022. Contains 356009 sequences. (Running on oeis4.)