A390397 Numbers whose sum of prime divisors equals the sum of square divisors.

12, 24, 63, 90, 189, 270, 304, 608, 950, 2107, 2640, 4018, 4750, 5280, 5312, 5775, 10624, 14749, 26378, 26533, 28126, 28875, 33615, 46893, 47236, 57420, 57477, 70380, 89586, 94472, 97767, 98112, 100845, 114840, 133133, 138591, 140760, 146510, 153738, 156087, 156638

Offset

1,1

Comments

Numbers k such that A008472(k) = A035316(k).

For every term t where prime p has an even multiplicity larger than 0 in the prime factorization of t we have t*p is in the sequence. - David A. Corneth, Dec 10 2025

Links

Felix Huber, Table of n, a(n) for n = 1..1410

Example

12 is a term 2 + 3 = 1 + 4.
63 is a term 3 + 7 = 1 + 9.
90 is a term 2 + 3 + 5 = 1 + 9.

Maple

filter:= proc(n) local t; add(t[1], t=ifactors(n)[2]) = add(select(issqr, NumberTheory:-Divisors(n))) end proc:
select(filter, [$1..156638]);

Mathematica

f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); q[1] = False; q[n_] := Module[{fct = FactorInteger[n]}, Times @@ f @@@ fct == Total[fct[[;; , 1]]]]; Select[Range[160000], q] (* Amiram Eldar, Dec 09 2025 *)

Prog

(PARI) isok(k) = my(f=factor(k)); vecsum(f[, 1]) == vecsum(select(issquare, divisors(f))); \\ Michel Marcus, Dec 09 2025

Crossrefs

Cf. A008472, A035316, A056240, A064502.

Sequence in context: A332037 A332039 A001335 * A206026 A145899 A172011

Adjacent sequences: A390394 A390395 A390396 * A390398 A390399 A390400

Keyword

nonn

Author

Felix Huber, Dec 09 2025

Status

approved