A211347 Numbers n such that n = sigma_k(m) for some k >= 1.

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 24, 26, 28, 30, 31, 32, 33, 36, 38, 39, 40, 42, 44, 48, 50, 54, 56, 57, 60, 62, 63, 65, 68, 72, 73, 74, 78, 80, 82, 84, 85, 90, 91, 93, 96, 98, 102, 104, 108, 110, 112, 114, 120, 121, 122

Offset: 1

Author

Jon Perry, Feb 05 2013

Keywords

• nonn

Comments

Sigma_k(n) = Sum[d|n, d^k].

Sigma_0(n) can be any positive integer and so is ignored in this sequence.

The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020

Examples

			Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.
• R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
• Eric W. Weisstein, MathWorld: Divisor function

Crossrefs

Cf. A000203, A001157, A001158, A001159, A001160.

Cf. A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972.

Cf. A000005.

Programs

Mathematica

upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1,Floor@Log[#,n]], #] & /@ Range[2,n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)

PARI

list(lim)=if(lim<3, return(if(lim<1,[],[1]))); my(v=List([1])); for(k=1,logint((lim\=1)-1,2), forfactored(m=2,sqrtnint(lim-1,k), my(t=sigma(m,k)); if(t<=lim, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Apr 09 2022