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A211347
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Numbers n such that n = sigma_k(m) for some k >= 1.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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
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EXAMPLE
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Sigma_2(4) = 1 + 4 + 16 = 21 so 21 is in the sequence.
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MATHEMATICA
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upto[n_] := Select[Union@Flatten[{1, DivisorSigma[Range@Max[1, Floor@Log[#, n]], #] & /@ Range[2, n]}], # <= n &]; upto[122] (* Giovanni Resta, Feb 05 2013 *)
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PROG
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(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
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CROSSREFS
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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.
Sequence in context: A304812 A026500 A091213 * A258777 A298004 A039238
Adjacent sequences: A211344 A211345 A211346 * A211348 A211349 A211350
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KEYWORD
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nonn
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AUTHOR
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Jon Perry, Feb 05 2013
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STATUS
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approved
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