A300664 Infinitary 3-nondeficient numbers: numbers k such that isigma(k) >= 3*k, where isigma(k) is the sum of infinitary divisors of k (A049417).

120, 840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3480, 3720, 4440, 4920, 5160, 5640, 5880, 6360, 7080, 7320, 7560, 8040, 8520, 8760, 9240, 9480, 9720, 9960, 10680, 10920, 11640, 11880, 12120, 12360

Offset

1,1

Comments

Analogous to 3-nondeficient numbers (A023197) with isigma (A049417) instead of sigma (A000203).

The infinitary 3-abundant numbers (analogous to A068403) are the terms that are not infinitary 3-perfect (120, 73440, 44553600, 252927360, 8402296320, 575409744000, ..., which are terms in A007358). - Amiram Eldar, Sep 16 2025

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

840 is in the sequence since isigma(840) = 2880 > 3 * 840.

Mathematica

f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[15000], isigma[#] >= 3*# &] (* Amiram Eldar, Sep 16 2025 *)

Prog

(PARI) isigma(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(f[i, 2]); prod(k = 1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1))); }
isok(k) = isigma(k) >= 3*k; \\ Amiram Eldar, Sep 16 2025

Crossrefs

Cf. A007357, A007358, A023197, A049417, A068403, A129656, A129657.

Sequence in context: A251264 A206025 A235551 * A029573 A211471 A250519

Adjacent sequences: A300661 A300662 A300663 * A300665 A300666 A300667

Keyword

nonn

Author

Amiram Eldar, Mar 10 2018

Extensions

Name corrected by Amiram Eldar, Sep 16 2025

Status

approved