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A336849
a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.
14
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, 17, 125, 33, 31, 35, 37, 243, 65, 57, 77, 225, 41, 23, 85, 189, 43, 55, 47, 9, 175, 29, 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
OFFSET
1,2
COMMENTS
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.
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
FORMULA
a(n) = A003961(n) / A336850(n) = A003961(n) / gcd(A003961(n), A003973(n)).
a(n) = A017666(A003961(n)).
MATHEMATICA
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 *)
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
\\ Or alternatively as:
A336849(n) = { my(u=A003961(n)); denominator(sigma(u)/u); };
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Antti Karttunen, Aug 06 2020
STATUS
approved