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