A364557 Möbius transform of A005941.

1, 1, 2, 2, 4, 2, 8, 4, 4, 4, 16, 4, 32, 8, 4, 8, 64, 4, 128, 8, 8, 16, 256, 8, 8, 32, 8, 16, 512, 4, 1024, 16, 16, 64, 8, 8, 2048, 128, 32, 16, 4096, 8, 8192, 32, 8, 256, 16384, 16, 16, 8, 64, 64, 32768, 8, 16, 32, 128, 512, 65536, 8, 131072, 1024, 16, 32, 32, 16, 262144, 128, 256, 8, 524288, 16, 1048576, 2048

Offset: 1

Author

Antti Karttunen, Jul 28 2023

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005941, A008683, A297112, A297113, A297167, A364558.

Programs

PARI

A364557(n) = if(1==n, 1, 2^(primepi(vecmax(factor(n)[, 1]))+(bigomega(n)-omega(n))-1));

PARI

A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
A364557(n) = sumdiv(n,d,moebius(n/d)*A005941(d));

Python

from sympy import factorint, primepi
def A364557(n): return 1<1 else 1 # Chai Wah Wu, Jul 29 2023

Formula

a(n) = Sum_{d|n} A008683(n/d) * A005941(d).

a(1) = 1; for n > 1, a(n) = A297112(n) = 2^(A297113(n)-1) = 2^A297167(n).