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A348047
a(n) = gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.
7
1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 8, 8, 30, 72, 32, 9, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 24, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32
OFFSET
1,2
COMMENTS
This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 196 = 4*49, where a(196) = 3, although a(4) = 1 and a(49) = 4.
FORMULA
a(n) = gcd(A000203(n), A003959(n)).
a(n) = gcd(A000203(n), A348029(n)) = gcd(A003959(n), A348029(n)).
a(n) = A000203(n)/ A348048(n) = A003959(n) / A348049(n).
MATHEMATICA
f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[Times @@ f @@@ FactorInteger[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Oct 21 2021 *)
PROG
(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348047(n) = gcd(sigma(n), A003959(n));
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 21 2021
STATUS
approved