A078346 a(1) = 1; a(n) = Sum_{k=1..n-1} a(floor((n-1)/k)).

1, 1, 2, 4, 7, 11, 17, 24, 34, 46, 62, 79, 104, 130, 163, 201, 249, 298, 363, 429, 513, 605, 714, 824, 966, 1112, 1284, 1468, 1687, 1907, 2181, 2456, 2779, 3120, 3510, 3910, 4394, 4879, 5430, 6008, 6677, 7347, 8139, 8932, 9836, 10788, 11850, 12913

Offset: 1

Author

Benoit Cloitre, Nov 22 2002

Keywords

• nonn

Crossrefs

Partial sums of A320224.

Programs

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A078346(n):
    if n == 1:
        return 1
    c, j, k1 = n, 1, n-1
    while k1 > 1:
        j2 = (n-1)//k1 + 1
        c += (j2-j)*A078346(k1)
        j, k1 = j2, (n-1)//j2
    return c-j # Chai Wah Wu, Apr 29 2025

Formula

For k>1, a(prime(k)+1)=2*a(prime(k))-a(prime(k)-1)+1. - Benoit Cloitre, Aug 29 2004

G.f. A(x) satisfies: A(x) = x + (x/(1 - x)) * Sum_{k>=1} (1 - x^k) * A(x^k). - Ilya Gutkovskiy, Aug 11 2021