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

%I A025523 #43 May 02 2025 02:52:45
%S A025523 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 A025523 93,97,105,106,119,120,136,139,142,145,171,172,175,178,198,199,212,
%U A025523 213,221,229,232,233,281,283,291,294,302,303,323,326,346,349,352,353,397,398,401
%N A025523 a(n) = 1 + Sum_{ k < n and k | n} a(k).
%C A025523 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 A025523 Partial sums of A074206, ordered factorizations. - _Augustine O. Munagi_, Jul 10 2007
%C A025523 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 A025523 Seiichi Manyama, <a href="/A025523/b025523.txt">Table of n, a(n) for n = 1..10000</a>
%H A025523 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 A025523 Running sum of A002033.
%F A025523 a(n) = 1 + Sum_{k = 2 to n} a([n/k]). - Mitchell Lee (worthawholephan(AT)gmail.com), Jul 26 2010
%F A025523 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
%t A025523 f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; Accumulate[Array[f, 100]] (* _Amiram Eldar_, May 02 2025 *)
%o A025523 (Python)
%o A025523 from functools import lru_cache
%o A025523 @lru_cache(maxsize=None)
%o A025523 def A025523(n):
%o A025523     if n == 0:
%o A025523         return 1
%o A025523     c, j = 2, 2
%o A025523     k1 = n//j
%o A025523     while k1 > 1:
%o A025523         j2 = n//k1 + 1
%o A025523         c += (j2-j)*A025523(k1)
%o A025523         j, k1 = j2, n//j2
%o A025523     return n+c-j # _Chai Wah Wu_, Mar 30 2021
%Y A025523 Cf. A002321, A022825, A347031.
%Y A025523 Cf. A074206, A002033.
%Y A025523 A173382 is an essentially identical sequence.
%K A025523 nonn
%O A025523 1,2
%A A025523 _David W. Wilson_