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A163500 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] exactly x times. 2
199981, 28263827, 371599983, 499999984, 10000000000, 9500000000, 9465000000, 9465000000, 10000000000 (list; graph; refs; listen; history; text; internal format)



This is an extension of a puzzle that a student posed as: Let f(x) be a function that counts how many times the digit 1 appears in the decimal representations of the numbers from 0 to x. So f(11) is 4. For what number > 1 does f(x) = x. The answer to that question is 199981, the first element of this sequence. The sequence is the natural extension of this property. Trivially s(0) doesn't exist, because for any x, [0...x] (inclusive) contains zero, meaning there is at least one matching substring, and this is a monotonically increasing function. It is not clear that s(n) is defined for all n>0, though the related sequence which uses f(x)>x rather than f(x)=x has at least less of a feeling of caprice about it. Multidigit n are clearly at a disadvantage, but I have tried to phrase it, "appears as a substring" so that, for example, 11 appears in 1111 thrice rather than twice.


Table of n, a(n) for n=1..9.


(Other) ;; this is in 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)) ;; this is just a progress indicator (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 f n) (display (matches-n-in-zero-to-k f n))) ;; where f should be one of = or > depending on which sequence you care about. ;; this could be made much more efficient, of course. In particular, the ;; initial sequences up to the first x of m digits have serious regularity.


See also A164321 which uses > instead of =. The first nine terms are contained in the sequences 1: A014778, 2: A101639, 3: A101640, 4:A101641, 5: A130427, 6: A130428, 7: A130429, 8: A130430, 9: A130431.

Sequence in context: A216400 A014778 A094799 * A164321 A230019 A106777

Adjacent sequences:  A163497 A163498 A163499 * A163501 A163502 A163503




Gregory Marton, Jul 29 2009, Aug 12 2009


a(5) - a(9) added by Gregory Marton, Aug 12 2009

Donovan Johnson pointed out the 6th term was incorrect, Nov 01 2010



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Last modified July 23 03:00 EDT 2019. Contains 325230 sequences. (Running on oeis4.)