A025523 a(n) = 1 + Sum_{ k < n and k | n} a(k).

1, 2, 3, 5, 6, 9, 10, 14, 16, 19, 20, 28, 29, 32, 35, 43, 44, 52, 53, 61, 64, 67, 68, 88, 90, 93, 97, 105, 106, 119, 120, 136, 139, 142, 145, 171, 172, 175, 178, 198, 199, 212, 213, 221, 229, 232, 233, 281, 283, 291, 294, 302, 303, 323, 326, 346, 349, 352, 353, 397, 398, 401

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Permanent of n X n (0,1) matrix defined by A(i,j)=1 iff j=1 or i divides j. - Vladeta Jovovic, Jul 05 2003

Partial sums of A074206, ordered factorizations. - Augustine O. Munagi, Jul 10 2007

The subsequence of primes begins: 2, 3, 5, 19, 29, 43, 53, 61, 67, 97, 139, 199, 229, 233, 281, 283, 349, 353, 397, 401. - Jonathan Vos Post

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics, Volume 11(2), 2004, R#10.

Crossrefs

Cf. A002321, A022825, A347031.

Cf. A074206, A002033.

A173382 is an essentially identical sequence.

Programs

Mathematica

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; Accumulate[Array[f, 100]] (* Amiram Eldar, May 02 2025 *)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A025523(n):
    if n == 0:
        return 1
    c, j = 2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A025523(k1)
        j, k1 = j2, n//j2
    return n+c-j # Chai Wah Wu, Mar 30 2021

Formula

Running sum of A002033.

a(n) = 1 + Sum_{k = 2 to n} a([n/k]). - Mitchell Lee (worthawholephan(AT)gmail.com), Jul 26 2010

G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x + Sum_{k>=2} (1 - x^k) * A(x^k)). - Ilya Gutkovskiy, Aug 11 2021