A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6144, 6272, 10976, 12500, 21600, 30375, 36000, 48600, 54675, 77760, 84375, 114688, 116640, 121500, 134456, 138240, 169344, 173056, 225000, 229376, 247808, 337500, 354294, 384000, 395136, 600000, 653184, 655360, 703125, 750141, 823543, 857304, 913952, 979776

Offset: 1

Author

Giuseppe Coppoletta, May 08 2016

Keywords

• nonn

Comments

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Moreover, due to the multiplicativity of the arithmetic function sigma, A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e., any permutation of its prime signature) gives also a term.

The condition defining this sequence coincides with the condition in A272858 at least for the terms of A114129.

Examples

			173056 is included because 173056 = 2^10 * 13^2 and sigma(2*13) = sigma(10*2).
653184 is included because 653184 = 2^7 * 3^6 * 7 and sigma(2*3*7) = sigma(7*6*1).
		

Links

• Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)

Crossrefs

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272858.

Programs

Mathematica

Select[Range[10^6], First@ # == Last@ # &@ Map[DivisorSigma[1, Times @@ #] &, Transpose@ FactorInteger@ #] &] (* Michael De Vlieger, May 12 2016 *)

Sage

A272859 = []
for n in (1..10000):
    v = factor(n)
    if prod(1 + w[0] for w in v) == sigma(prod(w[1] for w in v)): A272859.append(n)
print(A272859)