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A304203
If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).
2
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
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. A064988 (apply prime to p), A321874 (apply prime to both p & e).
Sequence in context: A354165 A073395 A064549 * A087687 A264090 A345283
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, May 09 2018
STATUS
approved