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a(n) = 1 + Sum_{ k < n and k | n} a(k).
10

%I #38 Mar 31 2023 14:07:00

%S 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,

%T 93,97,105,106,119,120,136,139,142,145,171,172,175,178,198,199,212,

%U 213,221,229,232,233,281,283,291,294,302,303,323,326,346,349,352,353,397,398,401

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

%C 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

%C Partial sums of A074206, ordered factorizations. - _Augustine O. Munagi_, Jul 10 2007

%C 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_

%H Seiichi Manyama, <a href="/A025523/b025523.txt">Table of n, a(n) for n = 1..10000</a>

%H Herbert S. Wilf, <a href="https://doi.org/10.37236/1867">The Redheffer matrix of a partially ordered set</a>, The Electronic Journal of Combinatorics, Volume 11(2), 2004, R#10.

%F Running sum of A002033.

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

%F 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

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A025523(n):

%o if n == 0:

%o return 1

%o c, j = 2, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c += (j2-j)*A025523(k1)

%o j, k1 = j2, n//j2

%o return n+c-j # _Chai Wah Wu_, Mar 30 2021

%Y Cf. A002321, A022825, A347031.

%Y Cf. A074206, A002033.

%Y A173382 is an essentially identical sequence.

%K nonn

%O 1,2

%A _David W. Wilson_