A164321 Let s(n) be the smallest number x such that the decimal representation of n appears as a substring of the decimal representations of the numbers [0...x] more than x times.

199991, 28263828, 371599993, 499999994, 5555555555, 6666666666, 7777777777, 8888888888, 9999999999

Offset: 1

Author

Gregory Marton, Aug 12 2009

Keywords

• nonn
• base
• more

Comments

a(10) <= 10^92 + 10^91 - 90. - Giovanni Resta, Aug 13 2019

Links

• Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 4.

Crossrefs

Closely related to A163500 substituting > for = as suggested by Alexey Radul. The first term is given in the related A092175 which also generalizes the sequence for bases other than 10.

Programs

Scheme

;; This is a program in PLT Scheme, a.k.a. mzscheme
(define (count-matches re str start-pos)
(let ((m (regexp-match-positions re str start-pos)))
(if m (+ 1 (count-matches re str (+ (caar m) 1))) 0)))
(define (matches-n-in-zero-to-k fn n)
(do ((sum-so-far 1)
(k (+ n 1))
(re (regexp (format "~a" n))))
((fn sum-so-far k) k)
(when (equal? 0 (modulo k 1000000))
(display (format "~a ~a ~a\n" n k sum-so-far)))
(set! k (+ k 1))
(set! sum-so-far
(+ sum-so-far (count-matches re (format "~a" k) 0)))))
(define (s n)
(matches-n-in-zero-to-k > n))