

A331737


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


3



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
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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} ((11/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 A097448
Adjacent sequences: A331734 A331735 A331736 * A331738 A331739 A331740


KEYWORD

nonn,mult,changed


AUTHOR

Antti Karttunen, Feb 02 2020


STATUS

approved



