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A336657
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Numbers k such that 2^k - 1 is divisible by the sum of the distinct primes dividing k (A008472).
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1
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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
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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12 = 2^2 * 3 is a term since 2^12 - 1 = 4095 is divisible by 2 + 3 = 5.
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MATHEMATICA
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b[n_] := Total[FactorInteger[n][[;; , 1]]]; Select[Range[2, 1500], PowerMod[2, #, b[#]] == 1 &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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