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%I #17 Dec 06 2021 03:05:51
%S 1,1,2,3,2,1,4,9,10,1,2,1,4,2,4,27,2,5,4,3,8,1,6,3,14,2,50,3,2,2,6,81,
%T 4,1,8,5,4,2,8,9,2,4,4,3,4,3,6,9,44,7,4,3,6,25,4,9,8,1,2,1,6,3,40,243,
%U 8,2,4,3,4,4,2,5,6,2,28,3,8,4,4,27,250,1,6,2,4,2,4,9,8,2,16,9,4,3,8,27,4,22,20
%N Numerators of the Möbius transform of ratio A003961(n)/n.
%C Because the ratio A003961(n)/n is multiplicative, so is also its Möbius transform. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349634 gives the denominators. See the examples.
%H Antti Karttunen, <a href="/A349633/b349633.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e The ratio a(n)/A349634(n) for n = 1..15: 1/1, 1/2, 2/3, 3/4, 2/5, 1/3, 4/7, 9/8, 10/9, 1/5, 2/11, 1/2, 4/13, 2/7, 4/15.
%t f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator @ DivisorSum[n, MoebiusMu[n/#] * s[#]/# &]; Array[a, 100] (* _Amiram Eldar_, Nov 28 2021 *)
%o (PARI)
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A349633(n) = numerator(sumdiv(n,d,moebius(n/d)*(A003961(d)/d)));
%Y Cf. A000010, A003961, A003972, A008683, A349634 (denominators).
%Y Cf. also A349627.
%K nonn,frac
%O 1,3
%A _Antti Karttunen_, Nov 28 2021
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