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A343878 a(n) is the least k such that A342585(k) = n.

%I A343878 #24 Oct 12 2021 21:54:27
%S A343878 1,2,5,9,11,21,25,30,47,39,59,71,96,100,126,115,160,178,197,217,221,
%T A343878 261,243,265,336,322,374,419,397,479,425,485,551,583,649,618,723,653,
%U A343878 801,690,727,887,930,974,889,932,1115,976,1260,1310,1023,1414,1070,1522
%N A343878 a(n) is the least k such that A342585(k) = n.
%C A343878 The term after the n-th 0 in A342585 is the running total of 0's, and there are infinitely many 0's, so all nonnegative integers appear in A342585. - _Peter Munn_, May 08 2021
%H A343878 Rémy Sigrist, <a href="/A343878/b343878.txt">Table of n, a(n) for n = 0..10000</a>
%H A343878 Rémy Sigrist, <a href="/A343878/a343878.gp.txt">PARI program for A343878</a>
%F A343878 For n >= 1, a(n) <= A343880(n) + 1. - _Peter Munn_, May 08 2021
%e A343878 We have:
%e A343878 n:       1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21
%e A343878 A342585: 0, 1, 1, 0, 2, 2, 2, 0, 3, 2,  4,  1,  1,  0,  4,  4,  4,  1,  4,  0,  5
%e A343878 So:
%e A343878 - a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 9, a(4) = 11, a(5) = 21.
%t A343878 Block[{a, c, k, m, nn = 54}, c[0] = 1; a = {0}~Join~Reap[Do[k = 0; While[IntegerQ[c[k]], Set[m, c[k]]; Sow[m]; If[IntegerQ@ c[m], c[m]++, c[m] = 1]; k++]; Sow[0]; c[0]++, nn]][[-1, -1]]; TakeWhile[Array[FirstPosition[a, #][[1]] &, nn, 0], IntegerQ]] (* _Michael De Vlieger_, Oct 12 2021 *)
%o A343878 (PARI) See Links section.
%o A343878 (Python)
%o A343878 def A343878(n):
%o A343878     k, c = 0, dict()
%o A343878     while True:
%o A343878         m, r = 0, 1
%o A343878         while r > 0:
%o A343878             k += 1
%o A343878             r = c.get(m,0)
%o A343878             if n == r:
%o A343878                 return k
%o A343878             c[r] = c.get(r,0)+1
%o A343878             m += 1 # _Chai Wah Wu_, Aug 31 2021
%Y A343878 Cf. A342585, A343880.
%Y A343878 For records see A347305.
%K A343878 nonn,look,easy
%O A343878 0,2
%A A343878 _Rémy Sigrist_, May 02 2021
%E A343878 Name shortened by _Peter Munn_, May 08 2021