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Numbers n such that n = sigma_k(m) for some k >= 1.
5

%I #31 Apr 09 2022 23:01:57

%S 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,

%T 38,39,40,42,44,48,50,54,56,57,60,62,63,65,68,72,73,74,78,80,82,84,85,

%U 90,91,93,96,98,102,104,108,110,112,114,120,121,122

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

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

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

%C The asymptotic density of this sequence is 0 (Niven, 1951, Rao and Murty, 1979). - _Amiram Eldar_, Jul 23 2020

%H Giovanni Resta, <a href="/A211347/b211347.txt">Table of n, a(n) for n = 1..10000</a>

%H Ivan Niven, <a href="https://doi.org/10.1090/S0002-9904-1951-09543-9">The asymptotic density of sequences</a>, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434.

%H R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, <a href="https://doi.org/10.4153/CMB-1979-018-5">On a theorem of Niven</a>, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/DivisorFunction.html">MathWorld: Divisor function</a>

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

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

%o (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

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

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

%Y Cf. A000005.

%K nonn

%O 1,2

%A _Jon Perry_, Feb 05 2013