A110079 Numbers k such that sigma(k) = 2k - 2^d(k) where d(k) = A000005(k) is the number of positive divisors of k.

5, 38, 284, 1370, 2168, 26828, 133088, 1515608, 19414448, 23521328, 25812848, 49353008, 82988756, 103575728, 537394688, 558504608, 921747488, 2651596448, 17517611968, 18249863488, 77792665408, 556915822208, 9311639470208, 12979901297948, 20006131221848, 29297682437888, 70644704198528

Offset

1,1

Comments

If 4^m+2^m-1 is prime (m in A098855), then k = 2^(m-1)*(4^m+2^m-1) is in the sequence because 2k-2^d(k) = 2^m*(4^m+2^m-1)-2^(m*2) = 2^m* (4^m-1) = 2^m*(2^m-1)*(2^m+1) = (2^m-1)*(4^m+2^m) = sigma(2^(m-1))*sigma(4^m+2^m-1) = sigma(2^(m-1)*(4^m+2^m-1)) = sigma(k). A110082 gives such terms of this sequence.

a(1) = 5 is the only odd (and prime) term. For n > 1, A007814(a(n)) >= A001222(a(n))-2. - Max Alekseyev, Jun 11 2026

Links

Max Alekseyev, Table of n, a(n) for n = 1..68

Mathematica

Do[If[DivisorSigma[1, n] == 2n - 2^DivisorSigma[0, n], Print[n]], {n, 925000000}]

Crossrefs

A110082 is a subsequence.

Cf. A000005, A098855, A125246, A125247, A125248, A191363, A275997, A292557, A363285, A387352.

Sequence in context: A129048 A279489 A027323 * A110082 A073508 A282964

Adjacent sequences: A110076 A110077 A110078 * A110080 A110081 A110082

Keyword

nonn,changed

Author

Farideh Firoozbakht, Aug 03 2005

Extensions

a(18)-a(21) from Donovan Johnson, Jan 31 2009

a(22) from Donovan Johnson confirmed by Giovanni Resta, Aug 14 2013

a(23) from Donovan Johnson confirmed and terms a(24) onward added by Max Alekseyev, Jun 11 2026

Status

approved