A348946 a(n) = gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 1, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 2, 56, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 4, 57, 93, 72, 98, 54, 6, 72, 24, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126, 96, 144, 72, 3, 74, 114, 124, 140, 96, 168, 80, 6, 1, 126

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 36 = 2^2 * 3^2, where a(36) = 1 <> 91 = 7*13 = a(4)*a(9).

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sigma(n)

Formula

a(n) = gcd(A000203(n), A348944(n)).

a(n) = gcd(A000203(n), A348945(n)) = gcd(A348944(n), A348945(n));

a(n) = A348944(n) / A348947(n) = A000203(n) / A348948(n).

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (f = FactorInteger[n]), (Times @@ f2 @@@ f + Times @@ f3 @@@ f)/2]; 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])); };
A348946(n) = gcd(sigma(n), ((1/2)*(A003959(n)+A034448(n))));

Crossrefs

Cf. A000203, A003959, A034448, A348944, A348945, A348947, A348948.

Cf. also A344695, A348047, A348503, A348733.

Sequence in context: A113957 A366148 A367171 * A366744 A073185 A284341

Adjacent sequences: A348943 A348944 A348945 * A348947 A348948 A348949

Keyword

nonn

Author

Antti Karttunen, Nov 05 2021

Status

approved