A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

1, 4, 16, 27, 72, 108, 432, 800, 3125, 6272, 12500, 21600, 30375, 50000, 84375, 121500, 169344, 225000, 247808, 337500, 486000, 750141, 823543, 1350000, 1384448, 3000564, 3294172, 6690816, 12002256, 13176688, 19600000, 22235661, 37380096, 37879808, 59295096, 88942644

Offset

1,2

Comments

Fixed points of A008477.

a(3) = 16 is the only term of the form p^q with p <> q. - Bernard Schott, Mar 28 2021

Links

Table of n, a(n) for n=1..36.

Example

16 = 2^4 = 4^2.
27 = 3^3.
108 = 2^2*3^3.
6272 = 2^7*7^2.
121500 = 2^2 * 3^5*5^3.

Mathematica

f[n_] := Product[{p, e} = pe; e^p, {pe, FactorInteger[n]}];
Reap[For[n = 1, n <= 10^8, n++, If[f[n] == n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 29 2021 *)

Prog

(PARI) for(n=2, 10^8, if(n==prod(i=1, omega(n), component(component(factor(n), 2), i)^component(component(factor(n), 1), i)), print1(n, ", ")))

Crossrefs

Cf. A000040, A008477.

Some subsequences: p_i^p_i (A051674), Product_i {p_i^p_i} (A048102), Product_(j,k)(p_j^p_k * p_k^p_j) with p_j < p_k (A082949) (see examples).

Sequence in context: A361078 A072653 A368107 * A201009 A111260 A067688

Adjacent sequences: A008475 A008476 A008477 * A008479 A008480 A008481

Keyword

nonn

Author

Olivier Gérard

Extensions

More terms from David W. Wilson

a(34)-a(36) from Jean-François Alcover, Mar 29 2021

Status

approved