A054412 Numbers k such that, in the prime factorization of k, 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

Author

Leroy Quet, May 09 2000

Keywords

• nonn

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

Examples

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

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A054412

Crossrefs

Cf. A051674, A054411, A082949.

Programs

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 *)

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.

Extensions

More terms from James Sellers, May 23 2000

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