A230593 a(n) = n * Sum_{q|n} 1 / q, where q are noncomposite numbers (A008578) dividing n.

1, 3, 4, 6, 6, 11, 8, 12, 12, 17, 12, 22, 14, 23, 23, 24, 18, 33, 20, 34, 31, 35, 24, 44, 30, 41, 36, 46, 30, 61, 32, 48, 47, 53, 47, 66, 38, 59, 55, 68, 42, 83, 44, 70, 69, 71, 48, 88, 56, 85, 71, 82, 54, 99, 71, 92, 79, 89, 60, 122, 62, 95, 93, 96, 83, 127

Offset: 1

Author

Jaroslav Krizek, Oct 25 2013

Keywords

• nonn

Examples

			For n = 6: a(6) = 6 * (1/1 + 1/2 + 1/3) = 11.
		

Links

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

Crossrefs

Cf. A080339, A000027, A008578, A329039.

Coincides with A129283 on squarefree numbers, A005117.

Programs

Mathematica

a[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; Array[a, 100] (* Amiram Eldar, Nov 12 2021 *)

PARI

A230593(n) = sumdiv(n,d,((1==d)||isprime(d))*(n/d)); \\ Antti Karttunen, Nov 12 2021

Formula

For n > 1, a(n) = n + n * Sum_(p|n) 1 / p, where p are primes dividing n.

a(n) = A069359(n) + n.

a(n) = Sum_{d|n} A080339(d) * A000027(n/d).

a(n) = A080339(n) * A000027(n), where operation * denotes Dirichlet convolution, i.e. convolution of type: a(n) = Sum_{d|n} b(d) * c(n/d).

For p, q = distinct primes, a(p) = p + 1, a(pq) = pq - 1.

From Antti Karttunen, Nov 12 2021: (Start)

a(n) = A129283(n) - A329039(n).

a(A005117(n)) = A129283(A005117(n)), for all n >= 1.

(End)

For p prime, k>=1, a(p^k) = p^(k-1) * (p+1). - Bernard Schott, Nov 12 2021