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

1,3

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.

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)

Example

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A204455 A318653 A161820 * A116528 A357111 A081431

Adjacent sequences: A341039 A341040 A341041 * A341043 A341044 A341045

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Feb 03 2021

Status

approved