A348733 a(n) = gcd(A003959(n), A034448(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

1, 3, 4, 1, 6, 12, 8, 9, 2, 18, 12, 4, 14, 24, 24, 1, 18, 6, 20, 6, 32, 36, 24, 36, 2, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 2, 38, 60, 56, 54, 42, 96, 44, 12, 12, 72, 48, 4, 2, 6, 72, 14, 54, 12, 72, 72, 80, 90, 60, 24, 62, 96, 16, 1, 84, 144, 68, 18, 96, 144, 72, 18, 74, 114, 8, 20, 96, 168, 80, 6, 2, 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 1444 = 2^2 * 19^2, where a(1444) = 10 != 1*2 = a(4)*a(361). See A348740 for the list of such positions.

Links

Antti Karttunen, Table of n, a(n) for n = 1..21125

Formula

a(n) = gcd(A003959(n), A034448(n)).

a(n) = gcd(A003959(n), A348732(n)) = gcd(A034448(n), A348732(n)).

a(n) = A003959(n) / A348734(n) = A034448(n) / A348735(n).

Mathematica

f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

Prog

(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348733(n) = gcd(A003959(n), A034448(n));

Crossrefs

Cf. A003959, A034448, A348732, A348734, A348735, A348740.

Cf. also A344695, A348047, A348503, A348946 for similar, almost multiplicative sequences.

Sequence in context: A347090 A328181 A358346 * A368471 A351569 A163762

Adjacent sequences: A348730 A348731 A348732 * A348734 A348735 A348736

Keyword

nonn

Author

Antti Karttunen, Nov 05 2021

Status

approved