A115853 -id:A115853 - cp's OEIS frontend

A171738 Number of n-digit terms in A115853.

0, 9, 9, 252, 819, 11754, 72585, 803448, 6978159, 73047510, 744922341, 8023947732, 88219609227, 993117723282, 11397388906305, 132852212160624, 1568346473860839, 18699577205645646, 224600363892164061, 2711096523623447820, 32815659723020049411, 397495008150096639114

Offset: 1

Zak Seidov, Dec 17 2009

Matthew House, Table of n, a(n) for n = 1..996
Index entries for linear recurrences with constant coefficients, signature (220, -23595, 1644214, -83716633, 3320655624, -106839372783, 2866614688938, -65446367283297, 1290934186719996, -22263630922575471, 338916526924263606, -4589394288200909781, 55636105851803049936, -607029027026607279171, 5987676299284026149946, -53597962913874542531511, 436793267723498418791916, -3249590798297591166114241, 22121531283842396531966386, -138066798468560248794910299, 791349936276906779789655496, -4171090447580882317650845341, 20240413531917813060282288558, -90503364704585154511917229755, 373146584463992545811390561028, -1419285864685331116742908160565, 4981481723295791035938228563970, -16135667263673226340044156972087, 48229932136237067328168799757664, -132993220541110610037595712319729, 338163168478885883461275641450094, -792363490141660083452341647812844, 1709441117632380221064502861863864, -3391977966130763237758648740886704, 6182428869770409717691044568257504, -10334792538707263177934951305545664, 15815751917789533624508526223437184, -22110429327513201841008428041853184, 28167208461596745245212310681342464, -32604111806052111009853487965004800, 34175732364859420044995467888502784, -32312338074096507084007090667556864, 27429278013461965544387449830678528, -20791237185298123019559840061734912, 13980827805208637058110391660085248, -8274715534505360843464786947735552, 4269275618057660740303251391905792, -1897199932751339113592691061948416, 715083586820263550683946460119040, -224022790650071250708428803276800, 56733883860143782469881036800000, -11154067599888012976481894400000, 1596567281510638039125196800000, -147945254464527104212992000000, 6658606584104736522240000000).

Cf. A115853 (numbers where every present decimal digit occurs more than once).

Table[9 Sum[k! Binomial[9, k] (-1)^i Binomial[n, i] StirlingS2[n - i, k - i + 1], {k, 0, 9}, {i, 0, Min[n, k + 1]}], {n, 21}] (* Matthew House, Sep 06 2020 *)

From Matthew House, Sep 13 2020: (Start)
a(n) = Sum_{k=0..9} k!*C(9,k)*(S_2(n,k) + k*S_2(n,k+1)) = 9*Sum_{k=0..9} k!*C(9,k)*S_2(n,k+1), where S_2(n,k) = A008299(n,k).
a(n) = 9*Sum_{k=0..10} (-1)^k*9!/(10-k)!*C(n,k)*(10-k)^(n-k) for n >= 10, where 0^0 = 1.
All terms from a(11) onward satisfy a linear recurrence with characteristic polynomial (1-x)^10*(2-x)^9*(3-x)^8*(4-x)^7*(5-x)^6*(6-x)^5*(7-x)^4*(8-x)^3*(9-x)^2*(10-x). (End)

More terms from Matthew House, Sep 06 2020

A033023 Numbers whose base-10 expansion has no run of digits with length < 2.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3300, 3311, 3322, 3333

Offset: 1

Clark Kimberling

Contains A014181 as subsequence. A115853 is a supersequence. - M. F. Hasler, Jun 24 2016

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1100 terms from Vincenzo Librandi)
Giovanni Resta, Zygodrome

Cf. A014181, A115853.

Select[Range[10000], Min[Length/@Split[IntegerDigits[#, 10]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

is(n)={n=digits(n);while(#n>2 && n[2]==n[1], n=if(n[3]==n[1],n[^1],n[3..-1]));#n>1&&n[1]==n[2]} \\ M. F. Hasler, Jun 24 2016

A320485 Keep just the digits of n that appear exactly once; write -1 if all digits disappear.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, -1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, -1, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, -1, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, -1, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, -1, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, -1, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, -1, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, -1, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, -1, 1, 0, 102, 103, 104, 105, 106, 107, 108, 109, 0, -1, 2, 3, 4, 5, 6, 7, 8, 9, 120, 2, 1, 123, 124, 125, 126, 127, 128, 129, 130, 3

Offset: 0

N. J. A. Sloane, Oct 24 2018

Digits that appear more than once in n are erased. Leading zeros are erased unless the result is 0. If all digits are erased, we write -1 for the result.
The map n -> a(n) was invented by Eric Angelini and described in a posting to the Sequence Fans Mailing List on Oct 24 2018.
More than the usual number of terms are shown in order to reach some interesting examples.
a(n) = -1 mostly. - David A. Corneth, Oct 24 2018

			1231 becomes 23, 1123 becomes 23, 11231 becomes 23, and 11023 becomes 23 (as we don't accept leading zeros). Note that 112323 disappears immediately and we get -1.
101 and 110 become 0 while 11000 and 10001 become -1.

Eric Angelini, Posting to Sequence Fans Mailing List, Oct 24 2018

Robert Israel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

See A320486 for another version.

Cf. A010784, A115853, A321008-A321012.

f:= proc(n) local F,S;
  F:= convert(n,base,10);
  S:= select(t -> numboccur(t,F)>1, [$0..9]);
  if S = {} then return n fi;
  F:= subs(seq(s=NULL,s=S),F);
  if F = [] then -1
  else add(F[i]*10^(i-1),i=1..nops(F))
  fi
end proc:
map(f, [$0..200]); # Robert Israel, Oct 24 2018

Array[If[# == {}, -1, FromDigits@ #] &@ Map[If[#[[-1]] > 1, -1, #[[1]] ] /. -1 -> Nothing &, Tally@ IntegerDigits[#]] &, 131] (* Michael De Vlieger, Oct 24 2018 *)

a(n) = {my(d=digits(n), v = vector(10), res = 0, t = 0); for(i=1, #d, v[d[i]+1]++); for(i=1, #d, if(v[d[i]+1]==1, t = 1; res=10 * res + d[i])); res - !t + !n} \\ David A. Corneth, Oct 24 2018

def A320485(n):
    return (lambda x: int(x) if x != '' else -1)(''.join(d if str(n).count(d) == 1 else '' for d in str(n))) # Chai Wah Wu, Nov 19 2018

From Rémy Sigrist, Oct 24 2018: (Start)
a(n) = n iff n belong to A010784.
a(n) <= 9876543210 with equality iff n = 9876543210.
(End)
If n > 9876543210, then a(n) < n. If a(n) < n, then a(n) <= 99n/1000. - Chai Wah Wu, Oct 24 2018

A350444 a(n) is the smallest number that has not appeared yet in the sequence and has only one digit in common with a(n-1).

Original entry on oeis.org

1, 10, 12, 2, 20, 21, 13, 3, 23, 24, 4, 14, 15, 5, 25, 26, 6, 16, 17, 7, 27, 28, 8, 18, 19, 9, 29, 32, 30, 31, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 67, 68, 69, 76, 70, 71, 72, 73, 74, 75, 78, 79, 87, 80, 81, 82, 83, 84, 85, 86, 89, 90

Offset: 1

Rodolfo Kurchan, Dec 31 2021

Terms computed by Claudio Meller.

It appears that this sequence contains all numbers except those in A115853.

			a(2) = 10 because it is the smallest number that has exactly one digit in common with a(1) = 1; similarly, a(3) = 12 because it has one digit in common with a(2) = 10 and a(4) = 2 because it is the smallest number that is not already in the sequence that has exactly one digit in common with a(3) = 12.

Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..10000 labeling the first 40 terms.

Cf. A115853, A350445.

c[] = False; j = c[1] = 1; {j}~Join~Reap[Do[d = Union@ IntegerDigits[j]; If[j == u, While[c[u] > 0, u++]]; k = u; While[Nand[c[k] == False, Count[IntegerDigits[k], ?(MemberQ[d, #] &)] == 1], k++]; Sow[k]; c[k] = True; j = k, 81]][[-1, -1]] (* Michael De Vlieger, Dec 31 2021 *)

from itertools import islice
def c(s, t): return sum(t.count(si) for si in s)
def agen(): # generator of terms
    an, target, seen, mink = 1, "1", {1}, 2
    while True:
        yield an
        k = mink
        while k in seen or c(str(k), target) != 1: k += 1
        an, target = k, str(k)
        seen.add(an)
        while mink in seen: mink += 1
print(list(islice(agen(), 82))) # Michael S. Branicky, Dec 31 2021

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

Original entry on oeis.org

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

Eric Angelini and Carole Dubois, Dec 17 2020

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)

			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.

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

Cf. base-10 Skolem-Langford numbers: A108116 (weak), A357826 (weaker), A132291 (strong).
Cf. A339611 (same idea turned into a different sequence).
Cf. A115853.

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

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

cp's OEIS Frontend

A171738 Number of n-digit terms in A115853.

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A033023 Numbers whose base-10 expansion has no run of digits with length < 2.

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A320485 Keep just the digits of n that appear exactly once; write -1 if all digits disappear.

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A350444 a(n) is the smallest number that has not appeared yet in the sequence and has only one digit in common with a(n-1).

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

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