A161946 Odd part of sum of unitary divisors of n.

1, 3, 1, 5, 3, 3, 1, 9, 5, 9, 3, 5, 7, 3, 3, 17, 9, 15, 5, 15, 1, 9, 3, 9, 13, 21, 7, 5, 15, 9, 1, 33, 3, 27, 3, 25, 19, 15, 7, 27, 21, 3, 11, 15, 15, 9, 3, 17, 25, 39, 9, 35, 27, 21, 9, 9, 5, 45, 15, 15, 31, 3, 5, 65, 21, 9, 17, 45, 3, 9, 9, 45, 37, 57, 13, 25, 3, 21, 5, 51, 41, 63, 21, 5, 27

Offset: 1

Author

Franklin T. Adams-Watters, Jun 22 2009

Keywords

• easy
• mult
• nonn

Comments

Since a(n) < n for any odd n > 1, iterating this function always leads to 1.

Links

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

Crossrefs

Cf. A000265, A034448, A161942, A151659.

Programs

Mathematica

Table[NestWhile[#/2 &, DivisorSum[n, # &, CoprimeQ[#, n/#] &], EvenQ@ # &], {n, 85}] (* Michael De Vlieger, Nov 23 2017 *)
a[n_] := Times @@ ((#/2^IntegerExponent[#, 2]) & /@ (1 + Power @@@ FactorInteger[n])); Array[a, 100] (* Amiram Eldar, Oct 26 2023 *)

PARI

oddpart(n)=n/2^valuation(n,2)
a(n)=local(fm,r);fm=factor(n);r=1;for(k=1,matsize(fm)[1],r*=oddpart(fm[k,1]^fm[k,2]+1));r

Formula

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

a(n) = A000265(A034448(n)). - Jason Yuen, Feb 02 2025