A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

1, 4, 9, 8, 25, 36, 49, 32, 27, 100, 121, 72, 169, 196, 225, 128, 289, 108, 361, 200, 441, 484, 529, 288, 125, 676, 243, 392, 841, 900, 961, 2048, 1089, 1156, 1225, 216, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 675, 2116, 2209, 1152, 343, 500, 2601, 1352, 2809, 972, 3025

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.

Index entries for sequences computed from exponents in factorization of n.

Formula

a(prime(i)^k) = prime(i)^prime(k).

a(A000040(k)) = A001248(k).

a(A001248(k)) = A030078(k).

a(A030078(k)) = A050997(k).

a(A002110(k)) = A061742(k).

Multiplicative with a(p^e) = p^prime(e). - M. F. Hasler, Nov 20 2018

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^prime(k)) = 1.80728269690724154161... . - Amiram Eldar, Jan 20 2024

Example

a(12) = a(2^2*3^1) = 2^prime(2)*3^prime(1) = 2^3*3^2 = 72.

Maple

a:= n-> mul(i[1]^ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 20 2021

Mathematica

a[n_] := Times @@ (#[[1]]^Prime[#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 55}]

Prog

(PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, f[k, 1]^prime(f[k, 2])); \\ Michel Marcus, May 09 2018
(PARI) apply( A304203(n)=factorback((n=factor(n))[, 1], apply(prime, n[, 2])), [1..50]) \\ M. F. Hasler, Nov 20 2018

Crossrefs

Cf. A000040, A001248, A002110, A003963, A008477, A030078, A048767, A050997, A056166 (records), A061742, A181819.

Cf. A064988 (apply prime to p), A321874 (apply prime to both p & e).

Sequence in context: A354165 A073395 A064549 * A087687 A264090 A345283

Adjacent sequences: A304200 A304201 A304202 * A304204 A304205 A304206

Keyword

nonn,mult

Author

Ilya Gutkovskiy, May 09 2018

Status

approved