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
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.
Programs
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Maple
with(numtheory): a:= n-> add(tau(tau (k)), k=1..n): seq(a(n), n=1..70); # Alois P. Heinz, Aug 28 2008
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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 *) -
PARI
a(n) = sum(k = 1, n, numdiv(numdiv(k))); \\ Michel Marcus, Sep 28 2013
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
Extensions
More terms from Alois P. Heinz, Aug 28 2008