A339803 Base-10 super-weak Skolem-Langford numbers.

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

Offset: 1

Author

Eric Angelini and Carole Dubois, Dec 17 2020

Keywords

• base
• nonn

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)

Examples

			a(1) = 2002: 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: 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.
		

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

Crossrefs

Cf. base-10 Skolem-Langford numbers: A108116 (weak), A357826 (weaker), A132291 (strong).

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

Cf. A115853.

Programs

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

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