A380760 Integers k with at least one proper factorization for which the sum of the same fixed integer power >= 2 of the factors equals k.

16, 27, 48, 54, 256, 270, 528, 1134, 1755, 2916, 3125, 7216, 7830, 11520, 11934, 15360, 19683, 22464, 30000, 31752, 40095, 40960, 46656, 65536, 69168, 81702, 86436, 93555, 100368, 146880, 200000, 212400, 264654, 273600, 291060, 303030, 317520, 340470, 362880

Offset

1,1

Comments

Superset of A381538 for values >= 16, and it is conjectured that the terms that match multiple nonequivalent factorizations here, such as a(5) = 256 (see Example), are exactly the terms of A381538 that can be produced as m^(m^e) by multiple m.

Links

Table of n, a(n) for n=1..39.

Example

a(1) = 16: 2 * 2 * 2 * 2 = 2^2 + 2^2 + 2^2 + 2^2 = 16.
a(5) = 256: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 = 256, and also 4 * 4 * 4 * 4 = 4^3 + 4^3 + 4^3 + 4^3 = 256.
a(8) = 1134: 2 * 3 * 3 * 7 * 9 = 2^3 + 3^3 + 3^3 + 7^3 + 9^3 = 1134.

Prog

(PARI) a380760_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, my(c=0); for(p=2, oo, my(t=sum(i=1, #f, f[i]^p)); if(t<x, next); if(t==x, c++); break); return(c)); my(d, c=0); fordiv(r, d, if(d==1 || d==x || (#f && d<f[#f]), next); listput(f, d); c+=a380760_count(x, f); listpop(f)); return(c)} \\ Charles L. Hohn, Mar 09 2025

Crossrefs

Sums of squares only: A380902.

Cf. A162247, A372053, A381538.

Sequence in context: A366962 A032610 A286429 * A380902 A329206 A384537

Adjacent sequences: A380757 A380758 A380759 * A380761 A380762 A380763

Keyword

nonn

Author

Charles L. Hohn, Feb 02 2025

Extensions

Edited by Peter Munn, Mar 25 2025

Status

approved