A161942 Odd part of sum of divisors of n.

1, 3, 1, 7, 3, 3, 1, 15, 13, 9, 3, 7, 7, 3, 3, 31, 9, 39, 5, 21, 1, 9, 3, 15, 31, 21, 5, 7, 15, 9, 1, 63, 3, 27, 3, 91, 19, 15, 7, 45, 21, 3, 11, 21, 39, 9, 3, 31, 57, 93, 9, 49, 27, 15, 9, 15, 5, 45, 15, 21, 31, 3, 13, 127, 21, 9, 17, 63, 3, 9, 9, 195, 37, 57, 31, 35, 3, 21, 5, 93, 121, 63

Offset: 1

Author

Franklin T. Adams-Watters, Jun 22 2009

Keywords

• easy
• mult
• nonn

Comments

It is conjectured that iteration of this function will always reach 1. This implies the nonexistence of odd perfect numbers. This is equivalent to the same question for A000593, which can be expressed as the sum of the divisors of the odd part of n.

Up to 20000000, there are only two odd numbers with a(n) and a(a(n)) both >= n: 81 and 18966025. See A162284.

For the nonexistence proof of odd perfect numbers, it is enough to show that this sequence has no fixed points beyond the initial one. This is equivalent to a similar condition given for A326042. - Antti Karttunen, Jun 17 2019

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sigma(n)
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000265, A000203, A000593, A162284, A326042, A336698. A348992, A351544.

Cf. A348741, A348742, A348743.

One less than A337194.

Programs

Mathematica

oddPart[n_] := n/2^IntegerExponent[n, 2]; a[n_] := oddPart[ DivisorSigma[1, n]]; Table[a[n], {n, 1, 82}] (* Jean-François Alcover, Sep 03 2012 *)

PARI

oddpart(n)=n/2^valuation(n,2);
a(n)=oddpart(sigma(n));

Python

from sympy import divisor_sigma
def A161942(n): return (m:=int(divisor_sigma(n)))>>(~m&m-1).bit_length() # Chai Wah Wu, Mar 17 2023

Scheme

(define (A161942 n) (A000265 (A000203 n))) ;; [For the implementations of A000203 and A000265, see under the respective entries]. - Antti Karttunen, Nov 18 2017

Formula

Multiplicative with a(p^e) = oddpart((p^{e+1}-1)/(p-1)), where oddpart(n) = A000265(n) is the largest odd divisor of n.

a(n) = A000265(A000203(n)).

a(n) = A337194(n)-1. - Antti Karttunen, Nov 30 2024