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A355933 a(n) = A003973(n) / gcd(sigma(n), A003973(n)), where A003973(n) = sigma(A003961(n)) and A003961 is fully multiplicative with a(p) = nextprime(p).

%I A355933 #13 Jul 23 2022 09:56:13
%S A355933 1,4,3,13,4,2,3,8,31,16,7,39,9,2,2,121,10,124,6,52,9,14,5,4,57,12,39,
%T A355933 39,16,8,19,52,7,40,2,31,21,8,27,32,22,3,12,13,124,5,9,363,7,76,5,117,
%U A355933 10,26,14,4,9,64,31,26,34,19,93,1093,12,7,18,130,15,8,37,248,40,28,171,78,7,18,21,484,71,88,15,117
%N A355933 a(n) = A003973(n) / gcd(sigma(n), A003973(n)), where A003973(n) = sigma(A003961(n)) and A003961 is fully multiplicative with a(p) = nextprime(p).
%C A355933 Numerator of ratio A003973(n) / A000203(n). This sequence gives the numerators when presented in its lowest terms, while A355934 gives the denominators. As both A000203 and A003973 are multiplicative sequences, their ratio is also: 1, 4/3, 3/2, 13/7, 4/3, 2/1, 3/2, 8/3, 31/13, 16/9, 7/6, 39/14, 9/7, 2/1, 2/1, 121/31, 10/9, 124/39, 6/5, etc.
%H A355933 Antti Karttunen, <a href="/A355933/b355933.txt">Table of n, a(n) for n = 1..20000</a>
%H A355933 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A355933 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A355933 a(n) = A003973(n) / A355932(n) = A003973(n) / gcd(A000203(n), A003973(n)).
%t A355933 f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* _Amiram Eldar_, Jul 22 2022 *)
%o A355933 (PARI)
%o A355933 A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
%o A355933 A355933(n) = { my(u=A003973(n)); (u/gcd(sigma(n), u)); };
%Y A355933 Cf. A000203, A003961, A003973, A355932, A355934 (denominators).
%Y A355933 Cf. also A341525, A349161.
%K A355933 nonn,frac
%O A355933 1,2
%A A355933 _Antti Karttunen_, Jul 22 2022