A226305 Numerator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5, 4095, 1, 17, 17, 65535, 1, 262143, 17, 65, 1, 4194303, 17, 69905, 1, 4161, 65, 268435455, 1, 1073741823, 257, 1025, 1, 53261, 13, 68719476735, 1, 4097, 257, 1099511627775, 1, 4398046511103, 1025, 3133, 1, 70368744177663, 257, 69810262081, 1, 65537, 4097

Offset: 1

Author

N. J. A. Sloane, Jun 07 2013

Keywords

• nonn
• frac

Examples

			1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5/3, 4095, 1, 17/3, 17, 65535, 1, 262143, 17/3, 65/3, 1, 4194303, 17/5, 69905, 1, 4161, 65/3, 268435455, 1, 1073741823, 257, 1025/3, 1, 53261/3, 13, ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• N. Bliss, B. Fulan, S. Lovett, and J. Sommars, Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials, Amer. Math. Monthly, 120 (2013), 519-536.

Crossrefs

Cf. A226306, A100371.

Programs

Maple

f:=proc(a,M) local n,b,d,t1,t2;
b:=[];
for n from 1 to M do
t1:=divisors(n);
t2:=mul(a[d]^mobius(n/d), d in t1);
b:=[op(b),t2];
od;
b;
end;
a:=[seq(2^phi(n)-1,n=1..100)];
f(a,100);

Mathematica

Table[Numerator[Product[(2^EulerPhi[d] - 1)^MoebiusMu[n/d], {d, Divisors[n]}]], {n, 100}] (* Indranil Ghosh, Apr 14 2017 *)

Python

from sympy import divisors, totient, mobius, prod
def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).numerator
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 14 2017