A139130 a(n) = Sum_{k=1..n} d(d(k)), where d(k) = number of divisors of k.

1, 3, 5, 7, 9, 12, 14, 17, 19, 22, 24, 28, 30, 33, 36, 38, 40, 44, 46, 50, 53, 56, 58, 62, 64, 67, 70, 74, 76, 80, 82, 86, 89, 92, 95, 98, 100, 103, 106, 110, 112, 116, 118, 122, 126, 129, 131, 135, 137, 141, 144, 148, 150, 154, 157, 161, 164, 167, 169, 175, 177, 180

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., Vol. 49, No. 2 (1948), 333-340. See Eq. 1.6. [From N. J. A. Sloane, Mar 12 2009]

Paul Erdős, On the sum Sum_{n=1..x} d[d(n)], Math. Student, Vol. 36 (1968), pp. 227-229.

E. Heppner, Über die Iteration von Teilerfunktionen, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.

Formula

a(n) = b * n * log(log(n)) + Sum_{k=0..floor(sqrt(n))} b_k * n/log(n)^k + O(n * exp(-c*sqrt(log(n)))), where b, b_k and c are constants (Heppner, 1974). - Amiram Eldar, Jan 15 2024

Maple

with(numtheory): a:= n-> add(tau(tau (k)), k=1..n): seq(a(n), n=1..70); # Alois P. Heinz, Aug 28 2008

Mathematica

Table[Sum[DivisorSigma[0, DivisorSigma[0, k]], {k, 1, n}], {n, 1, 62}] (* Geoffrey Critzer, Sep 28 2013 *)
Accumulate[Table[DivisorSigma[0, DivisorSigma[0, k]], {k, 1, 62}]] (* Amiram Eldar, Jan 15 2024 *)

Prog

(PARI) a(n) = sum(k = 1, n, numdiv(numdiv(k))); \\ Michel Marcus, Sep 28 2013

Crossrefs

Cf. A000005, A010553, A006218.

Sequence in context: A198082 A082767 A047932 * A219087 A186705 A361512

Adjacent sequences: A139127 A139128 A139129 * A139131 A139132 A139133

Keyword

nonn,easy

Author

Leroy Quet, Jun 05 2008

Extensions

More terms from Alois P. Heinz, Aug 28 2008

Status

approved