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%I #10 Apr 20 2023 10:34:21
%S 12,24,36,48,52,72,96,104,108,144,192,208,216,288,324,330,345,384,385,
%T 416,432,462,576,648,660,664,665,676,690,768,832,840,864,924,972,990,
%U 1035,1152,1190,1296,1302,1320,1328,1330,1352,1380,1386,1428,1430,1530,1536
%N Numbers k such that 2^k - 1 is divisible by the sum of the distinct primes dividing k (A008472).
%C 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.
%C The odd terms are relatively rare: 345, 385, 665, 1035, 1725, 1925, ...
%C 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.
%H Amiram Eldar, <a href="/A336657/b336657.txt">Table of n, a(n) for n = 1..10000</a>
%H William D. Banks and Florian Luca, <a href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=64fef7cb8bf93207659484ffcb8827e9a4d74849">Sums of prime divisors and Mersenne numbers</a>, Houston J. Math., Vol. 33, No. 2 (2007), pp. 403-413.
%F 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).
%e 12 = 2^2 * 3 is a term since 2^12 - 1 = 4095 is divisible by 2 + 3 = 5.
%t b[n_] := Total[FactorInteger[n][[;;, 1]]]; Select[Range[2, 1500], PowerMod[2, #, b[#]] == 1 &]
%Y Cf. A000225, A008472, A156787.
%K nonn
%O 1,1
%A _Amiram Eldar_, Jul 28 2020
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