A213931 Smallest number k such that the sum of divisors of k equals n times a nontrivial integer power.

3, 7, 6, 21, 19, 14, 12, 21, 22, 27, 43, 33, 63, 28, 24, 66, 67, 30, 98, 57, 44, 129, 367, 42, 199, 63, 85, 84, 463, 54, 48, 93, 86, 201, 76, 66, 219, 111, 99, 120, 163, 60, 1285, 129, 88, 274, 751, 105, 156, 199, 134, 198, 211, 102, 327, 84, 147, 346, 1765

Offset

1,1

Comments

Smallest k such that sigma(k) = n * m^q where m, q >= 2.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Example

a(34) = 201 because sigma(201) = 272 = 34*2^3.

Maple

with(numtheory):
a:= proc(n) local k, q;
      for k while irem(sigma(k), n, 'q')>0 or
      igcd(map(i->i[2], ifactors(q)[2])[])<2 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 26 2012

Mathematica

a[n_] := Module[{k, q, r}, For[k = 1, {q, r} = QuotientRemainder[ DivisorSigma[1, k], n]; r>0 || GCD @@ FactorInteger[q][[All, 2]]<2, k++]; k];
Array[a, 100] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)

Prog

(PARI) a(n)=my(k); while(sigma(k++)%n || !ispower(sigma(k)/n), ); k \\ Charles R Greathouse IV, Jun 26 2012

Crossrefs

Cf. A000203, A001597, A213401, A213386, A213420.

Sequence in context: A245602 A060924 A365100 * A213401 A363146 A013564

Adjacent sequences: A213928 A213929 A213930 * A213932 A213933 A213934

Keyword

nonn

Author

Michel Lagneau, Jun 25 2012

Status

approved