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

Author

Charles L. Hohn, Feb 02 2025

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Sums of squares only: A380902.

Cf. A162247, A372053, A381538.

Programs

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(tCharles L. Hohn, Mar 09 2025

Extensions

Edited by Peter Munn, Mar 25 2025