A293167 a(n) = Sum_{k = 1..n} d(d(d(k))), where d(k) is the number of divisors of k (A000005).

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 34, 37, 39, 42, 44, 46, 48, 51, 53, 55, 57, 60, 62, 65, 67, 70, 72, 74, 76, 78, 80, 82, 84, 87, 89, 92, 94, 97, 100, 102, 104, 107, 109, 112, 114, 117, 119, 122, 124, 127, 129, 131, 133, 137, 139, 141, 144, 146, 148, 151, 153, 156, 158

Offset: 1

Author

N. J. A. Sloane, Oct 17 2017

Keywords

• nonn

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340.
• Imre Kátai, On the iteration of the divisor-function, Publ. Math. Debrecen, Vol. 16 (1969), pp. 3-15.

Crossrefs

Part of the sequence A000005, A006218, A010553, A036450, A139130.

Programs

Mathematica

Accumulate[Table[DivisorSigma[0, DivisorSigma[0, DivisorSigma[0, n]]], {n, 80}]] (* Alonso del Arte, Oct 17 2017 *)

PARI

a(n) = sum(k=1, n, numdiv(numdiv(numdiv(k)))); \\ Michel Marcus, Oct 17 2017

PARI

first(n) = {my(v = vector(n)); v[1] = 1; for(i=2,n,v[i] = v[i-1] + numdiv(numdiv(numdiv(i)))); v} \\ David A. Corneth, Oct 17 2017

Formula

a(1) = 1; a(n + 1) = a(n) + A036450(n + 1) for n > 0. - David A. Corneth, Oct 17 2017

a(n) = (1 + o(1)) * c * n * log(log(log(n))), where c > 0 is a constant (Kátai, 1969). - Amiram Eldar, Apr 17 2024