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

1,2

Links

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

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A080339, A000027, A008578, A329039.

Coincides with A129283 on squarefree numbers, A005117.

Sequence in context: A206924 A185443 A275258 * A304411 A360522 A379497

Adjacent sequences: A230590 A230591 A230592 * A230594 A230595 A230596

Keyword

nonn

Author

Jaroslav Krizek, Oct 25 2013

Status

approved