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%I #42 Aug 10 2023 02:10:54
%S 2,4,8,4,21,5,45,49,83,10,269,11,202,412,479,15,1108,15,1545,1219,
%T 1343,21,8944,706,1043,5077,4084,28,27589,27,32160,10423,6689
%N a(n) is the number of positive integers k such that k is equal to the number of 1's in the digits of the base-n expansion of all positive integers <= k.
%C The greatest number counted by a(n) is 1...10, where the number of 1's is n-1. - Martin J. Erickson (erickson(AT)truman.edu), Oct 08 2010
%C These numbers, described in previous comment, 10(2), 110(3), 1110(4), ... expressed in base 10 are: 2, 12, 84, 780, 9330, 137256, 2396744, 48427560, 1111111110, ... - _Michel Marcus_, Aug 20 2013
%C The sequence described in the previous two comments is A226238. - _Ralf Stephan_, Aug 25 2013
%H Tanya Khovanova and Gregory Marton, <a href="https://arxiv.org/abs/2305.10357">Archive Labeling Sequences</a>, arXiv:2305.10357 [math.HO], 2023. See p. 9.
%e a(3)=4 since there are four values of k such that k is equal to the number of 1's in the digits of the base-3 expansion of all numbers <= k, namely, 1, 4, 5, 12.
%e From _Jon E. Schoenfield_, Apr 23 2023: (Start)
%e In the table below, an asterisk appears on each row k at which the cumulative count of 1's in the base-3 expansion of the positive integers 1..k is equal to k:
%e .
%e k #1's cume
%e ---------- ---- ----
%e 1 = 1_3 1 1*
%e 2 = 2_3 0 1
%e 3 = 10_3 1 2
%e 4 = 11_3 2 4*
%e 5 = 12_3 1 5*
%e 6 = 20_3 0 5
%e 7 = 21_3 1 6
%e 8 = 22_3 0 6
%e 9 = 100_3 1 7
%e 10 = 101_3 2 9
%e 11 = 102_3 1 10
%e 12 = 110_3 2 12*
%e (End)
%t nn = 7; Table[c = q = 0; Do[c += DigitCount[i, n, 1]; If[c == i, q++], {i, (#^# - #)/(# - 1) &[n]}]; q, {n, 2, nn}] (* _Michael De Vlieger_, May 24 2023 *)
%o (PARI) a(n) = {my(nmax = (n^n - 1)/(n - 1) - 1, s = 0, nb = 0); for (i=1, nmax, my(digs = digits(i, n)); s += sum (k=1, #digs, (digs[k] == 1)); if (s == i, nb++);); nb;} \\ _Michel Marcus_, Aug 20 2013; corrected Apr 23 2023
%Y Cf. A014778, A226238.
%K nonn,base,more
%O 2,1
%A Martin J. Erickson (erickson(AT)truman.edu), Sep 22 2009
%E Example corrected by Martin J. Erickson (erickson(AT)truman.edu), Sep 25 2009
%E Definition and a(10) corrected by _Tanya Khovanova_, Apr 23 2023
%E a(11)-a(35) from _Gregory Marton_, Jul 29 2023