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A348941
a(n) = n / gcd(n, A326042(n)).
5
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, 14, 29, 15, 31, 32, 33, 34, 35, 36, 37, 19, 39, 40, 41, 21, 43, 4, 45, 23, 47, 24, 49, 25, 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
OFFSET
1,2
COMMENTS
Denominator of ratio A326042(n) / n.
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.
FORMULA
a(n) = n / A348940(n) = n / gcd(n, A326042(n)).
MATHEMATICA
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 *)
PROG
(PARI)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A326042(n) = A064989(sigma(A003961(n)));
A348941(n) = (n / gcd(n, A326042(n)));
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Antti Karttunen, Nov 04 2021
STATUS
approved