A133482 a(p_1^e_1*p_2^e_2*.....*p_m^e_m) = (p_1^p_1)^e_1*(p_2^p^2)^e_2*.....*(p_m^p_m)^e_m where p_1^e_1*p_2^e_2*.....*p_m^e_m is the prime decomposition of n.

1, 4, 27, 16, 3125, 108, 823543, 64, 729, 12500, 285311670611, 432, 302875106592253, 3294172, 84375, 256, 827240261886336764177, 2916, 1978419655660313589123979, 50000, 22235661, 1141246682444, 20880467999847912034355032910567, 1728, 9765625, 1211500426369012, 19683, 13176688, 2567686153161211134561828214731016126483469, 337500

Offset: 1

Author

Masahiko Shin, Nov 29 2007

Keywords

• nonn
• mult

Comments

Totally multiplicative sequence with a(p) = p^p for prime p. - Jaroslav Krizek, Oct 17 2009

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^p - 1)) = 1.3850602852044891763... - Amiram Eldar, Dec 08 2020

Examples

			a(6) = a(2^1*3^1) = 2^2^1*3^3^1 = 4*27 = 108.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..388

Programs

Maple

A133482 := proc(n) local ifs,f ; if n = 1 then 1; else ifs := ifactors(n)[2] ; mul( (op(1,f)^op(1,f))^op(2,f), f=ifs) ; fi ; end: seq(A133482(n),n=1..30) ; # R. J. Mathar, Nov 30 2007

Mathematica

f[p_, e_] := (p^(p*e)); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 30] (* Amiram Eldar, Dec 08 2020 *)

Formula

Multiplicative with a(p^e) = p^(pe). If n = Product p(k)^e(k) then a(n) = Product p(k)^(p(k)*e(k)). - Jaroslav Krizek, Oct 17 2009

Extensions

Corrected and extended by R. J. Mathar, Nov 30 2007