A341042 Multiplicative projection of odd part of n.

1, 1, 3, 1, 5, 3, 7, 1, 6, 5, 11, 3, 13, 7, 15, 1, 17, 6, 19, 5, 21, 11, 23, 3, 10, 13, 9, 7, 29, 15, 31, 1, 33, 17, 35, 6, 37, 19, 39, 5, 41, 21, 43, 11, 30, 23, 47, 3, 14, 10, 51, 13, 53, 9, 55, 7, 57, 29, 59, 15, 61, 31, 42, 1, 65, 33, 67, 17, 69, 35, 71, 6, 73, 37, 30

Offset: 1

Author

Ilya Gutkovskiy, Feb 03 2021

Keywords

• nonn
• mult

Examples

			a(54) = a(2 * 3^3) = 3 * 3 = 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000026, A000079 (positions of 1's), A000265, A056911 (fixed points), A204455.

Programs

Maple

a:= n-> mul(`if`(i[1]=2, 1, i[1]*i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 03 2021

Mathematica

a[n_] := Times @@ (#[[1]] #[[2]] & /@ FactorInteger[n/2^IntegerExponent[n, 2]]); Table[a[n], {n, 75}]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1]==2, 1, f[i,1] * f[i,2]));} \\ Amiram Eldar, Nov 12 2022

Formula

a(n) = A000026(A000265(n)).

a(n) = A000026(n) if n odd, a(n) = a(n/2) if n even.

From Amiram Eldar, Nov 12 2022: (Start)

Multiplicative with a(2^e) = 1 and a(p^e) = e*p for p > 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (6*zeta(2)^2/17) * Product_{p prime} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.2947570019... . (End)