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A348941 a(n) = n / gcd(n, A326042(n)).

%I A348941 #19 Nov 15 2021 11:32:38
%S A348941 1,2,3,4,5,3,7,8,9,10,11,6,13,7,15,16,17,18,19,20,21,22,23,4,25,13,27,
%T A348941 14,29,15,31,32,33,34,35,36,37,19,39,40,41,21,43,4,45,23,47,24,49,25,
%U A348941 17,13,53,27,11,28,57,58,59,30,61,62,63,64,65,33,67,68,23,35,71,24,73,37,75,38,77,39,79,80,81,82
%N A348941 a(n) = n / gcd(n, A326042(n)).
%C A348941 Denominator of ratio A326042(n) / n.
%C A348941 If there are no more 1's in this sequence after the initial one, then there are no odd terms of A336702 (numbers whose abundancy index is a power of 2) larger than one, and neither there are odd terms in A005820 or in A046060. Compare to similar conditions given in A336848, A336849 and A337339.
%H A348941 Antti Karttunen, <a href="/A348941/b348941.txt">Table of n, a(n) for n = 1..16384</a>
%H A348941 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A348941 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348941 a(n) = n / A348940(n) = n / gcd(n, A326042(n)).
%t A348941 f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := n/GCD[n, s2[n]]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)
%o A348941 (PARI)
%o A348941 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A348941 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A348941 A326042(n) = A064989(sigma(A003961(n)));
%o A348941 A348941(n) = (n / gcd(n, A326042(n)));
%Y A348941 Cf. A005820, A046060, A326042, A336702, A336848, A336849, A337339, A348736, A348940, A348942 (numerators).
%K A348941 nonn,frac
%O A348941 1,2
%A A348941 _Antti Karttunen_, Nov 04 2021