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A022825 a(n) = a([ n/2 ]) + a([ n/3 ]) + . . . + a([ n/n ]) for n > 1, a(1) = 1.

%I A022825 #26 Feb 21 2022 09:53:01
%S A022825 1,1,2,3,4,6,7,9,11,13,14,19,20,22,25,29,30,36,37,42,45,47,48,60,62,
%T A022825 64,68,73,74,84,85,93,96,98,101,119,120,122,125,137,138,148,149,154,
%U A022825 162,164,165,193,195,201,204,209,210,226,229,241,244,246,247,278,279
%N A022825 a(n) = a([ n/2 ]) + a([ n/3 ]) + . . . + a([ n/n ]) for n > 1, a(1) = 1.
%H A022825 Ivan Neretin, <a href="/A022825/b022825.txt">Table of n, a(n) for n = 1..10000</a>
%F A022825 G.f. A(x) satisfies: A(x) = x + (1/(1 - x)) * Sum_{k>=2} (1 - x^k) * A(x^k). - _Ilya Gutkovskiy_, Feb 21 2022
%p A022825 a:= proc(n) option remember; `if`(n<2, 1,
%p A022825       add(a(iquo(n,j)), j=2..n))
%p A022825     end:
%p A022825 seq(a(n), n=1..63);  # _Alois P. Heinz_, Mar 31 2021
%t A022825 Fold[Append[#1, Total[#1[[Quotient[#2, Range[2, #2]]]]]] &, {1}, Range[2, 60]] (* _Ivan Neretin_, Aug 24 2016 *)
%o A022825 (Python)
%o A022825 from functools import lru_cache
%o A022825 @lru_cache(maxsize=None)
%o A022825 def A022825(n):
%o A022825     if n <= 1:
%o A022825         return n
%o A022825     c, j = 0, 2
%o A022825     k1 = n//j
%o A022825     while k1 > 1:
%o A022825         j2 = n//k1 + 1
%o A022825         c += (j2-j)*A022825(k1)
%o A022825         j, k1 = j2, n//j2
%o A022825     return c+n+1-j # _Chai Wah Wu_, Mar 31 2021
%Y A022825 Cf. A025523, A078346, A345182.
%K A022825 nonn
%O A022825 1,3
%A A022825 _Clark Kimberling_
%E A022825 Offset corrected by _Alois P. Heinz_, Mar 31 2021