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

%I A226305 #26 Jun 26 2025 14:03:03
%S A226305 1,1,3,3,15,1,63,5,21,1,1023,5,4095,1,17,17,65535,1,262143,17,65,1,
%T A226305 4194303,17,69905,1,4161,65,268435455,1,1073741823,257,1025,1,53261,
%U A226305 13,68719476735,1,4097,257,1099511627775,1,4398046511103,1025,3133,1,70368744177663,257,69810262081,1,65537,4097
%N A226305 Numerator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().
%H A226305 Vincenzo Librandi, <a href="/A226305/b226305.txt">Table of n, a(n) for n = 1..1000</a>
%H A226305 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 A226305 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 A226305 f:=proc(a,M) local n,b,d,t1,t2;
%p A226305 b:=[];
%p A226305 for n from 1 to M do
%p A226305 t1:=divisors(n);
%p A226305 t2:=mul(a[d]^mobius(n/d), d in t1);
%p A226305 b:=[op(b),t2];
%p A226305 od;
%p A226305 b;
%p A226305 end;
%p A226305 a:=[seq(2^phi(n)-1,n=1..100)];
%p A226305 f(a,100);
%t A226305 Table[Numerator[Product[(2^EulerPhi[d] - 1)^MoebiusMu[n/d], {d, Divisors[n]}]], {n, 100}] (* _Indranil Ghosh_, Apr 14 2017 *)
%o A226305 (Python)
%o A226305 from sympy import divisors, totient, mobius, prod
%o A226305 def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).numerator
%o A226305 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 14 2017
%Y A226305 Cf. A226306, A100371.
%K A226305 nonn,frac
%O A226305 1,3
%A A226305 _N. J. A. Sloane_, Jun 07 2013