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A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

%I A336849 #19 Feb 17 2021 20:26:06
%S A336849 1,3,5,9,7,5,11,27,25,21,13,15,17,11,35,81,19,75,23,63,55,39,29,9,49,
%T A336849 17,125,33,31,35,37,243,65,57,77,225,41,23,85,189,43,55,47,9,175,29,
%U A336849 53,135,121,49,19,17,59,125,13,99,115,93,61,105,67,111,275,729,119,65,71,171,29,77,73,135,79,41,245,69,143
%N A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.
%C A336849 If there are no more 1's in this sequence after the initial one, then there are no odd terms of A007691 (multiply perfect numbers) larger than one.
%C A336849 Denominator of the ratio A003973(n) / A003961(n), also denominator of the ratio (A341528(n)/A341529(n)) / (n / sigma(n)). - _Antti Karttunen_, Feb 16 2021
%H A336849 Antti Karttunen, <a href="/A336849/b336849.txt">Table of n, a(n) for n = 1..16383</a>
%H A336849 Antti Karttunen, <a href="/A336849/a336849.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A336849 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A336849 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A336849 a(n) = A003961(n) / A336850(n) = A003961(n) / gcd(A003961(n), A003973(n)).
%F A336849 a(n) = A017666(A003961(n)).
%t A336849 f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := (gn = g[n])/GCD[gn, DivisorSigma[1, gn]]; Array[a, 100] (* _Amiram Eldar_, Feb 17 2021 *)
%o A336849 (PARI)
%o A336849 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A336849 A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
%o A336849 \\ Or alternatively as:
%o A336849 A336849(n) = { my(u=A003961(n)); denominator(sigma(u)/u); };
%Y A336849 Cf. A003961, A003973, A007691, A017666, A336848, A336850, A341528, A341529, A341530.
%Y A336849 Cf. A341525 (numerators).
%K A336849 nonn,frac
%O A336849 1,2
%A A336849 _Antti Karttunen_, Aug 06 2020