A348984 a(n) = gcd(sigma(n), A325973(n)), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

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, 108

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 108 = 4*27, where a(108) = 4, although a(4) = 1 and a(27) = 8.

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), A325973(n)).

a(n) = gcd(A000203(n), A325974(n)) = gcd(A325973(n), A325974(n)).

a(n) = A000203(n) / A348985(n) = A325973(n) / A348986(n).

Mathematica

f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := GCD[s[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

Prog

(PARI)
A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348984(n) = gcd(sigma(n), A325973(n));

Crossrefs

Cf. A000203, A048250, A034448, A325973, A325974, A348985, A348986.

Differs from A348047 for the first time at n=108, where a(108) = 4, while A348047(108) = 8.

Cf. also A348733, A348946.

Sequence in context: A344695 A348503 A348047 * A323159 A347090 A328181

Adjacent sequences: A348981 A348982 A348983 * A348985 A348986 A348987

Keyword

nonn

Author

Antti Karttunen, Nov 06 2021

Status

approved