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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
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
Sequence in context: A355455 A059691 A097060 * A066085 A340511 A094529
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 28 2020
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)