A226305 - OEIS

%I #22 May 07 2020 16:30:52

%S 1,1,3,3,15,1,63,5,21,1,1023,5,4095,1,17,17,65535,1,262143,17,65,1,

%T 4194303,17,69905,1,4161,65,268435455,1,1073741823,257,1025,1,53261,

%U 13,68719476735,1,4097,257,1099511627775,1,4398046511103,1025,3133,1,70368744177663,257,69810262081,1,65537,4097

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

%H Vincenzo Librandi, <a href="/A226305/b226305.txt">Table of n, a(n) for n = 1..1000</a>

%H N. Bliss, B. Fulan, S. Lovett, and J. Sommars, <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.519">Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials</a>, Amer. Math. Monthly, 120 (2013), 519-536.

%e 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, ...

%p f:=proc(a,M) local n,b,d,t1,t2;

%p b:=[];

%p for n from 1 to M do

%p t1:=divisors(n);

%p t2:=mul(a[d]^mobius(n/d), d in t1);

%p b:=[op(b),t2];

%p od;

%p b;

%p end;

%p a:=[seq(2^phi(n)-1,n=1..100)];

%p f(a,100);

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

%o (Python)

%o from sympy import divisors, totient, mobius, prod

%o def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).numerator()

%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 14 2017

%Y Cf. A226306, A100371.

%K nonn,frac

%O 1,3

%A _N. J. A. Sloane_, Jun 07 2013