A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 24, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42, 85, 86, 87, 88, 89, 30

Offset

1,2

Comments

a(n) is the largest exponential divisor of n (cf. A322791) that is an exponentially odd number (A268335). - Amiram Eldar, Nov 17 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022).

Formula

a(n) = n / A331738(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p) * Sum_{k>=1} p^(2^k - 1)/(p^(2^(k+1)-2) - 1)) = 0.3953728204... . - Amiram Eldar, Nov 17 2022

Mathematica

f[p_, e_] := p^(e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

Prog

(PARI)
A000265(n) = (n>>valuation(n, 2));
A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };

Crossrefs

Cf. A000265, A268335, A322791, A331738.

Sequence in context: A333569 A110500 A161871 * A135875 A350390 A366765

Adjacent sequences: A331734 A331735 A331736 * A331738 A331739 A331740

Keyword

nonn,mult

Author

Antti Karttunen, Feb 02 2020

Status

approved