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A339803 Base 10 super-weak Skolem-Langford numbers. 2
2002, 30003, 131003, 200200, 231213, 300131, 312132, 400004, 420024, 1312132, 1410004, 2002000, 2002002, 2312131, 2312132, 3000300, 4000141, 5000005, 5300035, 12132003, 13100300, 14100141, 14130043, 15100005, 15120025, 20020000, 23121300, 23421314, 25121005, 25320035, 30003000, 30013100, 30023121, 31213200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Pick any digit d of a(n): there are exactly d digits between d and the closest duplicate of d (either before or after) inside a(n).

There are infinitely many such terms.

From M. F. Hasler, Dec 19 2020: (Start)

If N is a term of the sequence, then:

(1) Any digit of N must be present at least twice in N (Cf. A115853).

(2) N*10^k is also a term of the sequence, for all k >= 2.

(3) The reversal R(N) = A004086(N) is also a term (with leading zeros deleted). (End)

LINKS

David A. Corneth, Table of n, a(n) for n = 1..4080 (terms <= 10^12).

Kevin Ryde, Foma script creating DFA, and testing some properties

EXAMPLE

a(1) = 2002 and in 2002 the closest duplicate of the first 2 is 2 positions away to the right, the closest duplicate of the first 0 is 0 position away to the right, the closest duplicate of the second 0 is 0 position away to the left, the closest duplicate of the second 2 is 2 positions away to the left;

a(2) = 30003 and in 30003 the closest duplicate of the first 3 is 3 positions away to the right, the closest duplicate of the first 0 is 0 position away to the right, the closest duplicate of the second 0 is 0 position away (either to the left or to the right), the closest duplicate of the third 0 is 0 position away to the left, the closest duplicate of the second 3 is 3 positions away to the left;

a(13) = 2312131: if you pick any digit 1, the closest duplicate of this 1 is 1 position away (either to the left or to the right), if you pick any 2, the closest duplicate of this 2 is 2 positions away, if you pick any 3, the closest duplicate of this 3 is 3 positions away, etc.

PROG

(Python)

def nn(ti, t, s):

  li = s.rfind(t, 0, max(ti, 0))

  ri = s.find(t, min(ti+1, len(s)), len(s))

  if li==-1: li = -11

  if ri==-1: ri = len(s)+11

  return min(ti-li, ri-ti) - 1

def ok(n):

  strn = str(n)

  if any(strn.count(c)==1 for c in set(strn)): return False

  for i, c in enumerate(strn):

    if nn(i, c, strn) != int(c): return False

  return True

for n in range(6*10**6):

  if ok(n): print(n, end=", ") # Michael S. Branicky, Dec 17 2020

(PARI) is_A339803(n)={!for(i=1, #n=digits(n), (i>n[i]+1 && n[i-n[i]-1]==n[i])||(i+n[i]<#n && n[i+n[i]+1]==n[i])||return; for(j=max(i-n[i], 1), min(i+n[i], #n), n[j]==n[i] && j!=i && return))} \\ M. F. Hasler, Dec 19 2020

CROSSREFS

Cf. base-10 Skolem-Langford numbers: A108116 (weak), A132291 (strong).

Cf. A339611 (same idea turned into a different sequence).

Cf. A115853.

Sequence in context: A104400 A250880 A154049 * A108116 A140920 A162240

Adjacent sequences:  A339800 A339801 A339802 * A339804 A339805 A339806

KEYWORD

base,nonn

AUTHOR

Eric Angelini and Carole Dubois, Dec 17 2020

STATUS

approved

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Last modified January 29 05:09 EST 2022. Contains 350672 sequences. (Running on oeis4.)