cp's OEIS Frontend

A349627 Numerators of the Möbius transform of ratio A003961(n)/sigma(n).

%I A349627 #21 Dec 01 2021 20:16:36
%S A349627 1,0,1,2,1,0,3,18,35,0,1,1,3,0,1,126,1,0,3,1,3,0,5,9,77,0,125,3,1,0,5,
%T A349627 270,1,0,1,5,3,0,3,3,1,0,3,1,35,0,5,63,341,0,1,3,5,0,1,27,3,0,1,1,5,0,
%U A349627 105,1674,1,0,3,1,5,0,1,9,5,0,77,3,1,0,3,21,1975,0,5,3,1,0,1,3,7,0,9,5,5,0,1,135,3
%N A349627 Numerators of the Möbius transform of ratio A003961(n)/sigma(n).
%C A349627 Because the ratio A003961(n)/A000203(n) is multiplicative, so is also its Möbius transform. This sequence gives the numerator of that ratio when presented in its lowest terms, while A349628 gives the denominators. See the examples.
%H A349627 Antti Karttunen, <a href="/A349627/b349627.txt">Table of n, a(n) for n = 1..20000</a>
%H A349627 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349627 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A349627 The ratio a(n)/A349628(n) for n = 1..15: 1/1, 0/1, 1/4, 2/7, 1/6, 0/1, 3/8, 18/35, 35/52, 0/1, 1/12, 1/14, 3/14, 0/1, 1/24.
%t A349627 f[p_, e_] := NextPrime[p]^e*(p - 1)/(p^(e + 1) - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator @ DivisorSum[n, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349627 (PARI)
%o A349627 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A349627 A349627(n) = numerator(sumdiv(n,d,moebius(n/d)*(A003961(d)/sigma(d))));
%Y A349627 Cf. A000203, A003961.
%Y A349627 Cf. A349628 (denominators).
%Y A349627 Cf. also A342671, A349161, A349162, A349633.
%K A349627 nonn,frac
%O A349627 1,4
%A A349627 _Antti Karttunen_, Nov 26 2021