A348972 a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

1, 3, 4, 1, 6, 1, 8, 1, 1, 1, 12, 4, 14, 1, 1, 3, 18, 3, 20, 2, 1, 1, 24, 4, 1, 1, 2, 12, 30, 1, 32, 1, 1, 1, 1, 48, 38, 1, 1, 54, 42, 1, 44, 4, 12, 1, 48, 4, 1, 1, 1, 18, 54, 3, 1, 4, 1, 1, 60, 8, 62, 1, 2, 1, 1, 1, 68, 2, 1, 3, 72, 12, 74, 1, 2, 12, 1, 1, 80, 2, 1, 1, 84, 16, 1, 1, 1, 12, 90, 3, 1, 4, 1, 1, 1, 4, 98

Offset: 1

Author

Antti Karttunen, Nov 06 2021

Keywords

• nonn

Crossrefs

Cf. A003415, A003959, A129283, A348970, A348973, A348974.

Cf. also A343226.

Programs

Mathematica

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n])), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348972(n) = gcd(A003959(n),(n+A003415(n)));

Formula

a(n) = gcd(A003959(n), A129283(n)) = gcd(A003959(n), n+A003415(n)).

a(n) = gcd(A003959(n), A348970(n)) = gcd(A129283(n), A348970(n)).

a(n) = A129283(n) / A348973(n) = A003959(n) / A348974(n).