OFFSET
2,1
COMMENTS
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
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
The sequence described in the previous two comments is A226238. - Ralf Stephan, Aug 25 2013
LINKS
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 9.
EXAMPLE
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.
From Jon E. Schoenfield, Apr 23 2023: (Start)
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:
.
k #1's cume
---------- ---- ----
1 = 1_3 1 1*
2 = 2_3 0 1
3 = 10_3 1 2
4 = 11_3 2 4*
5 = 12_3 1 5*
6 = 20_3 0 5
7 = 21_3 1 6
8 = 22_3 0 6
9 = 100_3 1 7
10 = 101_3 2 9
11 = 102_3 1 10
12 = 110_3 2 12*
(End)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Martin J. Erickson (erickson(AT)truman.edu), Sep 22 2009
EXTENSIONS
Example corrected by Martin J. Erickson (erickson(AT)truman.edu), Sep 25 2009
Definition and a(10) corrected by Tanya Khovanova, Apr 23 2023
a(11)-a(35) from Gregory Marton, Jul 29 2023
STATUS
approved