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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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%I #37 Jan 11 2023 06:39:33

%S 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 124,127,128,168,186,192,217,224,248,254,256,336,372,381,384,434,448,

%U 496,508,512,651,672,744,762,768,868,889,896,992,1016,1024,1302,1344,1488

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

%C 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 The reverse is also true. All numbers in this sequence have this form. - _Ivan Neretin_, Aug 12 2016

%C From _Antti Karttunen_, Sep 01 2021: (Start)

%C Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.

%C Numbers k whose odd part [A000265(k)] is in A046528.

%C (End)

%H Ivan Neretin, <a href="/A054784/b054784.txt">Table of n, a(n) for n = 1..10000</a>

%F Numbers n such that A000203(2*n) - A000203(n) = 2^w for some w.

%F Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} (1 + 1/p) = 2 * A306204 = 3.1711177758... . - _Amiram Eldar_, Jan 11 2023

%e 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 N:= 10^6: # to get all terms <= N

%p M:= select(isprime, [seq(2^i-1, i=select(isprime, [$2..ilog2(N+1)]))]):

%p R:= map(t -> seq(2^i*t, i=0..floor(log[2](N/t))), map(convert,combinat:-powerset(M),`*`)):

%p sort(convert(R,list)); # _Robert Israel_, Aug 12 2016

%t Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2 - 1, PrimeQ]], # <= Max@p2 &] (* _Ivan Neretin_, Aug 12 2016 *)

%t Select[Range[1500],IntegerQ[Log2[DivisorSigma[1,2#]-DivisorSigma[1,#]]]&] (* _Harvey P. Dale_, Apr 23 2019 *)

%o (PARI)

%o A209229(n) = (n && !bitand(n,n-1));

%o isA054784(n) = A209229(sigma(n>>valuation(n,2))); \\ _Antti Karttunen_, Aug 28 2021

%Y Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).

%Y Positions of zeros in A336922. Positions of 0's and 1's in A336361.

%Y Cf. also A003401.

%K nonn

%O 1,2

%A _Labos Elemer_, May 22 2000