A328132 Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

300, 2100, 3300, 3900, 5100, 5700, 6900, 8700, 9300, 11100, 12100, 12300, 12900, 14100, 15900, 17700, 18300, 20100, 21300, 21900, 23100, 23700, 23760, 24900, 26700, 27300, 29100, 30300, 30900, 32100, 32700, 33900, 35700, 38100, 39300, 39900, 41100, 41700, 42900

Offset: 1

Author

Amiram Eldar, Oct 04 2019

Keywords

• nonn

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• J. Hanumanthachari, V. V. Subrahmanya Sastri, and V. Srinivasan, On e-perfect numbers, Math. Student, Vol. 46, No. 1 (1978), pp. 71-80; entire issue.

Crossrefs

The exponential version of A019281.

Cf. A051377, A064012.

Programs

Mathematica

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 3n; Select[Range[50000], espQ]

PARI

esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
isok(k) = esigma(esigma(k)) == 3*k; \\ Amiram Eldar, Jan 09 2025

Formula

300 is in the sequence since esigma(300) = 540, and esigma(540) = 900 = 3*300.