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A341270 a(n) = Sum_{k=1..n} a(n mod k) for n > 0; a(0) = 1.

%I A341270 #13 Feb 09 2021 03:54:17
%S A341270 1,1,2,3,4,6,7,10,12,15,17,25,24,32,37,45,46,63,62,82,83,97,104,141,
%T A341270 130,158,170,201,202,255,242,302,306,350,367,448,416,503,522,610,597,
%U A341270 716,690,825,832,921,945,1147,1085,1255,1272,1430,1435,1683,1631,1888
%N A341270 a(n) = Sum_{k=1..n} a(n mod k) for n > 0; a(0) = 1.
%e A341270 a(1) = a(1 mod 1) = a(0) = 1.
%e A341270 a(2) = a(2 mod 1)+a(2 mod 2) = a(0)+a(0) = 2.
%e A341270 a(3) = a(3 mod 1)+a(3 mod 2)+a(3 mod 3) = a(0)+a(1)+a(0) = 3.
%p A341270 a:= proc(n) option remember;
%p A341270      `if`(n=0, 1, add(a(n mod k), k=1..n))
%p A341270     end:
%p A341270 seq(a(n), n=0..62);  # _Alois P. Heinz_, Feb 07 2021
%t A341270 a[0] = 1; a[n_] := a[n] = Sum[a[Mod[n, k]], {k, 1, n}]; Array[a, 50, 0] (* _Amiram Eldar_, Feb 08 2021 *)
%o A341270 (Python)
%o A341270 a = [1]
%o A341270 for n in range(1,1000):
%o A341270     a.append(sum(a[n%k] for k in range(1,n+1)))
%o A341270 (PARI) a(n) = if (n==0, 1, sum(k=1, n, a(n % k))); \\ _Michel Marcus_, Feb 08 2021
%Y A341270 For Sum_{k=1..n} n mod k see A004125.
%Y A341270 For Sum_{k=1..n} a(k) see A000079.
%Y A341270 For Max_{k=1..n} a(n mod k)+1 see A113473.
%K A341270 nonn
%O A341270 0,3
%A A341270 _Rok Cestnik_, Feb 07 2021