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A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

%I A054784 #37 Jan 11 2023 06:39:33
%S A054784 1,2,3,4,6,7,8,12,14,16,21,24,28,31,32,42,48,56,62,64,84,93,96,112,
%T A054784 124,127,128,168,186,192,217,224,248,254,256,336,372,381,384,434,448,
%U A054784 496,508,512,651,672,744,762,768,868,889,896,992,1016,1024,1302,1344,1488
%N A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.
%C A054784 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.
%C A054784 The reverse is also true. All numbers in this sequence have this form. - _Ivan Neretin_, Aug 12 2016
%C A054784 From _Antti Karttunen_, Sep 01 2021: (Start)
%C A054784 Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.
%C A054784 Numbers k whose odd part [A000265(k)] is in A046528.
%C A054784 (End)
%H A054784 Ivan Neretin, <a href="/A054784/b054784.txt">Table of n, a(n) for n = 1..10000</a>
%F A054784 Numbers n such that A000203(2*n) - A000203(n) = 2^w for some w.
%F A054784 Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} (1 + 1/p) = 2 * A306204 = 3.1711177758... . - _Amiram Eldar_, Jan 11 2023
%e A054784 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
%p A054784 N:= 10^6: # to get all terms <= N
%p A054784 M:= select(isprime, [seq(2^i-1, i=select(isprime, [$2..ilog2(N+1)]))]):
%p A054784 R:= map(t -> seq(2^i*t, i=0..floor(log[2](N/t))), map(convert,combinat:-powerset(M),`*`)):
%p A054784 sort(convert(R,list)); # _Robert Israel_, Aug 12 2016
%t A054784 Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2 - 1, PrimeQ]], # <= Max@p2 &] (* _Ivan Neretin_, Aug 12 2016 *)
%t A054784 Select[Range[1500],IntegerQ[Log2[DivisorSigma[1,2#]-DivisorSigma[1,#]]]&] (* _Harvey P. Dale_, Apr 23 2019 *)
%o A054784 (PARI)
%o A054784 A209229(n) = (n && !bitand(n,n-1));
%o A054784 isA054784(n) = A209229(sigma(n>>valuation(n,2))); \\ _Antti Karttunen_, Aug 28 2021
%Y A054784 Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).
%Y A054784 Positions of zeros in A336922. Positions of 0's and 1's in A336361.
%Y A054784 Cf. also A003401.
%K A054784 nonn
%O A054784 1,2
%A A054784 _Labos Elemer_, May 22 2000