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

1,3

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)

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)

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 *)

Prog

(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<<primepi(p)-1)<<i for i, p in enumerate(factorint(n, multiple=True)))) if n > 1 else 0 # Chai Wah Wu, Mar 10 2023

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.

Sequence in context: A258052 A344484 A241448 * A346193 A244974 A077580

Adjacent sequences: A323240 A323241 A323242 * A323244 A323245 A323246

Keyword

nonn

Author

Antti Karttunen, Jan 10 2019

Status

approved