A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

0, 1, 3, 4, 7, 6, 15, 8, 12, 13, 31, 12, 63, 18, 18, 24, 127, 14, 255, 20, 39, 48, 511, 24, 28, 84, 24, 48, 1023, 32, 2047, 32, 54, 176, 42, 40, 4095, 258, 144, 56, 8191, 38, 16383, 68, 36, 800, 32767, 48, 60, 31, 252, 132, 65535, 30, 91, 72, 528, 1302, 131071, 44, 262143, 2736, 60, 104, 126, 96, 524287, 304, 774, 42, 1048575, 72, 2097151, 4356, 42

Offset: 1

Author

Antti Karttunen, Jan 10 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A156552, A323244, A323247, A323248, A324118, A324543 (Möbius transform), A324396, A324823.

Cf. A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Programs

Mathematica

Array[If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 75] (* Michael De Vlieger, Apr 21 2019 *)

PARI

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A323243(n) = if(1==n, 0, sigma(A156552(n)));

PARI

\\ For computing terms a(n), with n > ~4000 use Hans Havermann's factorization file https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt"); \\ First read it in as a PARI-vector.
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1))); \\ Then play sigma
\\ Antti Karttunen, Mar 15 2019

Python

from sympy import divisor_sigma, primepi, factorint
def A323243(n): return divisor_sigma(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

Formula

a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

a(n) = 2*A156552(n) - A323244(n).

a(n) = A323247(n) - A323248(n).

From Antti Karttunen, Mar 12 2019: (Start)

a(A000040(n)) = A000225(n).

a(n) = Sum_{d|n} A324543(d).

For n > 1, a(2*A246277(n)) = A324118(n).

gcd(a(n), A156552(n)) = A324396(n).

A000035(a(n)) = A324823(n).

(End)