A328133 Exponential (2,4)-perfect numbers: numbers m such that esigma(esigma(m)) = 4m, where esigma(m) is the sum of exponential divisors of m (A051377).

540, 3780, 5940, 7020, 9180, 10260, 12420, 15660, 16740, 19980, 22140, 23220, 25380, 28620, 31860, 32940, 36180, 38340, 39420, 41580, 42660, 44820, 48060, 49140, 52380, 54540, 55620, 57780, 58860, 61020, 64260, 68580, 70740, 71820, 73980, 75060, 77220, 80460

Offset

1,1

Comments

Conjecturally, a subsequence of A083207 (tested for the first 659 terms of this sequence). - Ivan N. Ianakiev, Oct 05 2019

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.

Example

540 is in the sequence since esigma(540) = 900, and esigma(900) = 2160 = 4*540.

Mathematica

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 4n; Select[Range[80000], espQ]

Prog

(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)) == 4*k; \\ Amiram Eldar, Jan 09 2025

Crossrefs

The exponential version of A019282.

Cf. A051377, A083207.

Sequence in context: A069474 A253824 A218414 * A146123 A159207 A101705

Adjacent sequences: A328130 A328131 A328132 * A328134 A328135 A328136

Keyword

nonn

Author

Amiram Eldar, Oct 04 2019

Status

approved