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A054784
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Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.
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9
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1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 31, 32, 42, 48, 56, 62, 64, 84, 93, 96, 112, 124, 127, 128, 168, 186, 192, 217, 224, 248, 254, 256, 336, 372, 381, 384, 434, 448, 496, 508, 512, 651, 672, 744, 762, 768, 868, 889, 896, 992, 1016, 1024, 1302, 1344, 1488
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OFFSET
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1,2
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COMMENTS
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If n is a squarefree product of Mersenne primes multiplied by a power of 2, then sigma(2n) - sigma(n) is a power of 2.
The reverse is also true. All numbers in this sequence have this form. - Ivan Neretin, Aug 12 2016
Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.
(End)
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LINKS
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FORMULA
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EXAMPLE
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For n=12, sigma(2n) = sigma(24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and sigma(n) = sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. So sigma(2n) - sigma(n) = 60 - 28 = 32 = 2^5 is a power of 2, and therefore 12 is in the sequence. - Michael B. Porter, Aug 15 2016
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MAPLE
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N:= 10^6: # to get all terms <= N
M:= select(isprime, [seq(2^i-1, i=select(isprime, [$2..ilog2(N+1)]))]):
R:= map(t -> seq(2^i*t, i=0..floor(log[2](N/t))), map(convert, combinat:-powerset(M), `*`)):
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MATHEMATICA
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Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2 - 1, PrimeQ]], # <= Max@p2 &] (* Ivan Neretin, Aug 12 2016 *)
Select[Range[1500], IntegerQ[Log2[DivisorSigma[1, 2#]-DivisorSigma[1, #]]]&] (* Harvey P. Dale, Apr 23 2019 *)
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PROG
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(PARI)
A209229(n) = (n && !bitand(n, n-1));
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CROSSREFS
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Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).
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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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