A324105 a(1) = 0; for n > 1, a(n) = A000005(A156552(n)).

0, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 2, 4, 4, 7, 2, 8, 2, 6, 4, 9, 2, 6, 4, 4, 4, 10, 4, 11, 2, 4, 4, 6, 4, 12, 2, 8, 4, 13, 2, 14, 2, 4, 8, 15, 2, 8, 3, 8, 2, 16, 2, 9, 2, 8, 6, 17, 2, 18, 4, 4, 6, 6, 4, 19, 4, 4, 2, 20, 4, 21, 4, 4, 4, 8, 4, 22, 2, 8, 4, 23, 6, 12, 8, 16, 8, 24, 6, 12, 4, 12, 12, 12, 4, 25, 3, 8, 4, 26, 6, 27, 2, 8

Offset: 1

Author

Antti Karttunen, Feb 18 2019

Keywords

• nonn

Comments

If a(n) is odd, then A067029(n) = 1 (and A319710(n) = 0), but not necessarily vice versa. See also formulas in A106737.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..4473
• Index entries for sequences related to binary expansion of n
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000005, A067029, A106737, A156552, A319710, A324051.

Cf. also A323243, A324104, A324119, A324117 for similarly permuted sigma, phi, omega and A001227 (number of odd divisors).

Programs

Mathematica

Array[If[# == 1, 0, DivisorSigma[0, #] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]] &, 105] (* Michael De Vlieger, Mar 11 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))));
A324105(n) = if(1==n,0,numdiv(A156552(n)));

Formula

a(1) = 0; for n > 1, a(n) = A000005(A156552(n)).