A332881 - OEIS

%I #14 Nov 15 2021 13:01:56

%S 1,2,3,2,5,1,7,2,3,5,11,1,13,7,5,2,17,1,19,5,21,11,23,1,5,13,3,7,29,5,

%T 31,2,11,17,35,1,37,19,39,5,41,7,43,11,5,23,47,1,7,5,17,13,53,1,55,7,

%U 57,29,59,5,61,31,21,2,65,11,67,17,23,35

%N If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

%C Denominator of sum of reciprocals of squarefree divisors of n.

%H Antti Karttunen, <a href="/A332881/b332881.txt">Table of n, a(n) for n = 1..20000</a>

%F Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).

%F a(n) = denominator of Sum_{d|n} mu(d)^2/d.

%F a(n) = denominator of psi(n)/n.

%F a(p) = p, where p is prime.

%F a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - _Antti Karttunen_, Nov 15 2021

%e 1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

%p a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 28 2020

%t Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator

%t Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator

%o (PARI)

%o A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615

%o A332881(n) = denominator(A001615(n)/n);

%Y Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.

%K nonn,frac

%O 1,2

%A _Ilya Gutkovskiy_, Feb 28 2020