A348942 - OEIS

%I #20 Nov 15 2021 11:33:02

%S 1,1,2,11,1,1,2,3,29,1,5,11,4,1,2,49,3,29,2,11,4,5,6,1,34,2,22,11,1,1,

%T 17,55,10,3,2,319,10,1,8,3,7,2,2,5,29,3,8,49,85,17,2,11,6,11,1,3,4,1,

%U 29,11,13,17,58,1091,4,5,4,33,4,1,31,29,3,5,68,11,10,4,10,49,469,7,12,11,3,1,2,15,25,29,8

%N a(n) = A326042(n) / gcd(n, A326042(n)).

%C Numerator of ratio A326042(n)/n. Ratio A326042(n)/n is multiplicative because both A326042 and A000027 are.

%H Antti Karttunen, <a href="/A348942/b348942.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A326042(n) / A348940(n) = A326042(n) / gcd(n, A326042(n)).

%F For all n >= 1, A348943(A064989(n)) = 1.

%t 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_] := (sn = s2[n])/GCD[n, sn]; Array[a, 100] (* _Amiram Eldar_, Nov 05 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 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 A326042(n) = A064989(sigma(A003961(n)));

%o A348942(n) = { my(u=A326042(n)); (u / gcd(n, u)); };

%Y Cf. A000203, A003961, A005820, A007691, A046060, A064989, A326042, A348739, A348940, A348941 (denominators), A348943.

%K nonn,frac

%O 1,3

%A _Antti Karttunen_, Nov 04 2021