A335497 a(1) = 1, and for any n > 0, a(n+1) is the number of times the decimal representation of a(n) appears in the concatenation of the first n terms, possibly with overlap.

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 12, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 4, 3, 3, 4, 4, 5, 3, 5, 4, 6, 3, 6, 4, 7, 3, 7, 4, 8, 3, 8, 4, 9, 3, 9, 4, 10, 3, 10, 4, 11, 5, 5, 6, 5, 7, 5, 8, 5, 9, 5, 10, 5, 11, 6, 6

Offset: 1

Author

Rémy Sigrist, Jun 14 2020

Keywords

• nonn
• base

Comments

This sequence is a variant of A276457.

This sequence is unbounded.

It seems that lim sup a(n)/(n*log(n)) = 0.03 approximately. - Ya-Ping Lu, Dec 16 2021

Examples

			The first terms, alongside their concatenations with a star in front of each occurrence of a(n), are:
  n   a(n)  cat(a(1)...a(n))
  --  ----  ---------------------------------
   1     1  *1
   2     1  *1*1
   3     2  11*2
   4     1  *1*12*1
   5     3  1121*3
   6     1  *1*12*13*1
   7     4  112131*4
   8     1  *1*12*13*14*1
   9     5  11213141*5
  ...
  17     9  1121314151617181*9
  18     1  *1*12*13*14*15*16*17*18*19*1
  19    10  112131415161718191*10
  20     1  *1*12*13*14*15*16*17*18*19*1*10*1
  21    12  1*12131415161718191101*12
  22     2  11*21314151617181911011*2*2
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Logarithmic scatterplot of the first 1000000 terms
• Rémy Sigrist, Perl program for A335497

Crossrefs

Cf. A179801, A276457.

Programs

Perl

See Links section.

Python

a1 = 1; print(a1, end =', '); S = str(a1)
for n in range(2, 100): ct = S.count(str(a1)); S += str(ct); print(ct, end = ', '); a1 = ct # Ya-Ping Lu, Dec 16 2021