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A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

%I A332883 #19 Feb 16 2025 08:33:59
%S A332883 1,2,3,4,5,1,7,8,9,5,11,3,13,7,5,16,17,3,19,2,21,11,23,2,25,13,27,7,
%T A332883 29,5,31,32,11,17,35,18,37,19,39,20,41,7,43,11,3,23,47,12,49,25,17,26,
%U A332883 53,9,55,7,57,29,59,1,61,31,63,64,65,11,67,34,23,35
%N A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).
%C A332883 Denominator of sum of reciprocals of unitary divisors of n.
%H A332883 Antti Karttunen, <a href="/A332883/b332883.txt">Table of n, a(n) for n = 1..20000</a>
%H A332883 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>
%F A332883 a(n) = denominator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
%F A332883 a(n) = denominator of usigma(n)/n.
%F A332883 a(p) = p, where p is prime.
%F A332883 a(n) = n / A323166(n). - _Antti Karttunen_, Nov 13 2021
%e A332883 1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...
%p A332883 a:= n-> denom(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
%p A332883 seq(a(n), n=1..80);  # _Alois P. Heinz_, Feb 28 2020
%t A332883 Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
%t A332883 Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Denominator
%o A332883 (PARI) a(n) = denominator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ _Michel Marcus_, Feb 28 2020
%Y A332883 Cf. A007947, A017666, A034448, A077610, A319677, A323166, A327158 (positions of 1's), A332881, A332882 (numerators).
%K A332883 nonn,frac
%O A332883 1,2
%A A332883 _Ilya Gutkovskiy_, Feb 28 2020