A054412 Numbers n such that, in the prime factorization of n, the product of exponents equals the product of prime factors.

1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6272, 12500, 21600, 30375, 36000, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 395136, 600000, 653184, 750141, 823543, 857304, 979776, 1384448, 1474560, 1500000

Offset

1,2

Comments

For p prime, numbers of the form p^p satisfy the condition, hence A051674 is a subsequence. - Michel Marcus, May 19 2014

Also, numbers of the form p^q * q^p, with distinct primes p and q, satisfy the condition, hence A082949 is a subsequence. - Bernard Schott, Apr 11 2020

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A054412

Example

192 is included because 192 =2^6 *3^1 and 2*3 = 6*1.

Mathematica

peppfQ[n_]:=Module[{f=Transpose[FactorInteger[n]]}, Times@@First[f] == Times@@Last[f]]; Select[Range[1.5*10^6], peppfQ] (* Harvey P. Dale, Oct 14 2015 *)

Prog

(PARI) isok(n) = my(f = factor(n)); prod(i=1, #f~, f[i, 2]) == prod(i=1, #f~, f[i, 1]); \\ Michel Marcus, May 19 2014
(PARI) See Links section.

Crossrefs

Cf. A051674, A054411, A082949.

Sequence in context: A272818 A356433 A272859 * A122405 A122406 A276372

Adjacent sequences: A054409 A054410 A054411 * A054413 A054414 A054415

Keyword

nonn

Author

Leroy Quet, May 09 2000

Extensions

More terms from James A. Sellers, May 23 2000

New name and three more terms from Michel Marcus, May 19 2014

Status

approved