A161942 - OEIS

%I #33 Nov 30 2024 23:46:39

%S 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,

%T 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,

%U 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

%N Odd part of sum of divisors of n.

%C 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.

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

%C 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

%H Antti Karttunen, <a href="/A161942/b161942.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

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

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

%F a(n) = A337194(n)-1. - _Antti Karttunen_, Nov 30 2024

%t 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 *)

%o (PARI)

%o oddpart(n)=n/2^valuation(n,2);

%o a(n)=oddpart(sigma(n));

%o (Scheme) (define (A161942 n) (A000265 (A000203 n))) ;; [For the implementations of A000203 and A000265, see under the respective entries]. - _Antti Karttunen_, Nov 18 2017

%o (Python)

%o from sympy import divisor_sigma

%o def A161942(n): return (m:=int(divisor_sigma(n)))>>(~m&m-1).bit_length() # _Chai Wah Wu_, Mar 17 2023

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

%Y Cf. A348741, A348742, A348743.

%Y One less than A337194.

%K easy,mult,nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Jun 22 2009