A032981 -id:A032981 - cp's OEIS frontend

A178403 Numbers containing the rounded up arithmetic mean of their digits at least once, cf. A004427.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 133, 134, 135, 143, 145, 146, 147, 153, 154, 157, 158, 159, 164, 169, 174, 175, 185

Offset: 1

Reinhard Zumkeller, May 27 2010

A178401(a(n)) > 0; complement of A178402.
A010785, A050278, A178358, A178359 are subsequences;
a(n) = A131207(n) for n < 48;
a(n) = A134336(n) for n < 48;
a(n+1) = A032981(n) for n < 38.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A134336 Nonnegative integers n containing each digit between n's smallest and largest decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 201, 210, 211, 212, 213, 221, 222, 223, 231, 232, 233, 234, 243, 312, 321, 322, 323

Offset: 1

Rick L. Shepherd, Oct 21 2007

A032981 is a subsequence; the term 102 is the first positive integer not also in A032981. A171102 (pandigital numbers) and A033075 are subsequences. Union of A010785 (repdigits) and A108965.
a(n) = A178403(n) for n < 48. - Reinhard Zumkeller, May 27 2010
Equivalently, numbers with the property that the set of its decimal digits is a set of consecutive numbers. - Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A032981, A050278, A033075 (a subsequence), A010785, A108965.

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8));for(i=2,#v,if(v[i]!=1+v[i-1],return(0)));1 \\ Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

is_A134336(n)={vecmax(n=Set(digits(n)))-vecmin(n)==#n-1} \\ M. F. Hasler, Dec 24 2014

def ok(n): d = sorted(set(map(int, str(n)))); return d[-1]-d[0]+1 == len(d)
print([k for k in range(324) if ok(k)]) # Michael S. Branicky, Dec 12 2023

a(n) ~ n. - Charles R Greathouse IV, Sep 09 2011

Edited by N. J. A. Sloane, Aug 06 2012

A235163 Number of positive integers with n digits in which adjacent digits differ by at most 1.

Original entry on oeis.org

9, 26, 75, 217, 629, 1826, 5307, 15438, 44941, 130900, 381444, 1111926, 3242224, 9455987, 27583372, 80472698, 234799873, 685149328, 1999414181, 5835044495, 17029601028, 49702671494, 145066398937, 423412132499, 1235854038791, 3607255734629, 10529101874491

Offset: 1

Gerry Leversha, Jan 04 2014

			a(2) = 26: 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 98, 99.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,1,6,-1).

Cf. A032981, A090994, A126364 (allowing leading zeros).

u:= proc(n, r) option remember; `if`(n=1, `if`(r=0, 0, 1),
      add(`if`(r+i in [$0..9], u(n-1, r+i), 0), i=-1..1))
    end:
a:= n-> add(u(n, r), r = 0..9):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 12 2014

CoefficientList[Series[-x*(3*x^4-18*x^3-9*x^2+28*x-9)/(x^5-6*x^4-x^3+10*x^2-6*x+1),{x,0,30}],x]//Rest (* Harvey P. Dale, Aug 13 2019 *)

from functools import cache
@cache
def u(n, r):
    if r < 0 or r > 9: return 0
    if n == 1: return (r > 0)
    return u(n-1, r-1) + u(n-1, r) + u(n-1, r+1)
def a(n): return sum(u(n, r) for r in range(10))
print([a(n) for n in range(1, 28)]) # Michael S. Branicky, Sep 26 2021

a(n) = Sum_{r=0..9} u(n,r) where u(n,r) = 0 if r<0 or r>9, u(1,0) = 0, u(1,r) = 1 for 1<=r<=9, and otherwise u(n,r) = u(n-1,r-1) + u(n-1,r) + u(n-1,r+1).

G.f.: -x*(3*x^4-18*x^3-9*x^2+28*x-9)/(x^5-6*x^4-x^3+10*x^2-6*x+1). - Alois P. Heinz, Jan 12 2014

A068148 Primes in which neighboring digits differ at most by 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 43, 67, 89, 101, 109, 211, 223, 233, 433, 443, 677, 787, 877, 887, 1009, 1109, 1123, 1223, 2111, 2221, 2333, 3221, 3323, 3343, 3433, 4567, 5443, 7789, 7877, 8887, 8999, 9001, 9011, 9887, 9901, 10009, 10099, 10111, 10909, 10987, 12101, 12109

Offset: 1

Amarnath Murthy, Feb 23 2002

Neighbors of 9 are 0 and 8 and 9.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A010051, subsequence of A032981.

a068148 n = a068148_list !! (n-1)
a068148_list = filter ((== 1) . a010051') a032981_list
-- Reinhard Zumkeller, Feb 14 2015

Do[a = IntegerDigits[ Prime[n]]; k = 1; l = Length[a]; While[k < l && (Abs[a[[k]]- a[[k + 1]]] < 2 || Abs[a[[k]] - a[[k + 1]]] > 8), k++ ]; If[k == l, Print[ Prime[n]]], {n, 1, 10^4} ]
Select[Prime[Range[1500]],Max[Abs[Differences[IntegerDigits[#]]]/.{9->1}] < 2&] (* Harvey P. Dale, Jan 31 2012 *)

Edited and extended by Robert G. Wilson v and Sascha Kurz, Mar 01 2002

Corrected by T. D. Noe, Feb 15 2008

A252490 Numbers whose set of digits is simply connected, with 9 and 0 considered as neighbors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 90, 98, 99, 100, 101, 102, 109, 110, 111, 112, 120, 121, 122, 123, 132, 190, 201, 210, 211, 212, 213, 221, 222, 223, 231, 232, 233, 234, 243, 312, 321, 322, 323, 324, 332, 333, 334, 342, 343, 344, 345, 354, 423, 432

Offset: 1

M. F. Hasler, Dec 24 2014

The set of digits must consist of a single run without "holes", but for a cyclic topology where 9 and 0 are seen as neighbors.

A superset of A134336. Namely, numbers in A134336 or such that the complement of their digits in {0,...,9} satisfies the criterion of A134336.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A032981, A050278, A033075 (a subsequence), A010785, A108965, A134336 (a subsequence).

is(n)=vecmax(if((d=Set(digits(n)))[1],d,d=setminus(vector(9,i,i),d)))-vecmin(d)==#d-1

A032987 Numbers with the property that all pairs of consecutive base-10 digits differ by more than 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 25, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 41, 47, 48, 49, 50, 51, 52, 58, 59, 60, 61, 62, 63, 69, 70, 71, 72, 73, 74, 80, 81, 82, 83, 84, 85, 90, 91, 92, 93, 94, 95, 96, 140, 141, 147, 148

Offset: 1

Clark Kimberling

Cf. A032981.

Select[Range[200],And@@(Abs[First[#]-Last[#]]>2&/@Partition[ IntegerDigits[#],2,1])&] (* Harvey P. Dale, Mar 28 2011 *)
Select[Range[200],Min[Abs[Differences[IntegerDigits[#]]]]>2&] (* Harvey P. Dale, May 24 2015 *)

A252481 Numbers whose set of digits is simply connected to 1.

Original entry on oeis.org

1, 10, 11, 12, 21, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 132, 201, 210, 211, 212, 213, 221, 231, 312, 321, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1023, 1032, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1123, 1132, 1200, 1201, 1202, 1203, 1210, 1211, 1212, 1213, 1220, 1221, 1222, 1223

Offset: 1

M. F. Hasler, Dec 24 2014

"Simply connected" means that there must not be a "hole" in the set of digits. E.g., {1,2,4} would not be allowed since '3' is missing.

Cf. A032981, A050278, A033075, A010785, A108965, A134336, A252490.

is(n)={d=Set(digits(n));d[1]==1 || (#d>1&&d[2]==1) || return; d[#d]==#d-!d[1]}

def ok(n):
  s = set(str(n)); return '1' in s and len(s) == ord(max(s))-ord(min(s))+1
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(1223)) # Michael S. Branicky, Jan 10 2021

A284046 Numbers k, not ending in 0, such that the consecutive digits of k^2 differ by 0 or 1.

Original entry on oeis.org

1, 2, 3, 11, 26, 111, 1111, 11111, 105462, 111111, 460688, 753576, 1111111, 2806538, 3513626, 5858612, 11111111, 23335688, 111111111, 674874474, 8226042716, 2131535935501, 81655720279388

Offset: 1

Giovanni Resta, Mar 19 2017

Equivalently, numbers not ending in 0, whose square belong to A032981.
All members k ending in 1 are generators of infinite numbers of the form k*10^e which satisfy the same property. In a sense, here we list only "primitive" terms, not ending in 0.
a(24) > 10^17, if it exists.

			81655720279388 belongs to this sequence because the consecutive digits of its square, 6667656654345656676777654544, differ by 0 or 1.

Cf. A032981, A048412, A079036.

Select[Range[10^6], Mod[#, 10] > 0 && Max@ Abs@ Differences@ IntegerDigits[ #^2] <= 1 &]

A357142 Nonnegative numbers all of whose pairs of consecutive decimal digits are adjacent digits, where 9 and 0 are considered adjacent.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 90, 98, 101, 109, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876, 878, 890, 898, 901, 909

Offset: 1

Ofer Zivony, Sep 14 2022

This is very similar to A033075, with the exception of considering 0 and 9 as adjacent digits. This allows these digits to be equal to the other digits, making it a more balanced list.

Cf. A032981, A033075, A043089 (ternary analog), A048491.

q:= n-> (l-> andmap(x-> x in {1, 9}, {seq(abs(l[i]-l[i-1]),
             i=2..nops(l))}))(convert(n, base, 10)):
select(q, [$0..1000])[];  # Alois P. Heinz, Sep 14 2022

q[n_] := AllTrue[Abs @ Differences @ IntegerDigits[n], MemberQ[{1, 9}, #] &]; Select[Range[0, 1000], q] (* Amiram Eldar, Sep 15 2022 *)

a(n) = { n--; for (b=0, oo, if (n <= 9*2^b, my (v=ceil(n/2^b), p=(n-1)%(2^b)); while (b>0, v=10*v+vecsort([(v-1)%10, (v+1)%10])[1+bittest(p,b--)];); return (v), n -= 9*2^b)) } \\ Rémy Sigrist, Sep 15 2022

def add_dig(x):
  d = (x%10-1)%8 if x%10 != 0 else 1
  return 10*x+d
def try_incr(x):
  if x < 10: return x+1
  r = x//10
  d2 = r%10
  d = max((d2+1)%10,(d2-1)%10)
  return 10*r+d
def incr(x):
  new_x=try_incr(x)
  return new_x if new_x>x else add_dig(incr(x//10))
x = 0
for n in range(1,1000):
  print(f"{n} {x}")
  x = incr(x)

cp's OEIS Frontend

A178403 Numbers containing the rounded up arithmetic mean of their digits at least once, cf. A004427.

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A134336 Nonnegative integers n containing each digit between n's smallest and largest decimal digits.

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A235163 Number of positive integers with n digits in which adjacent digits differ by at most 1.

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Maple

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A068148 Primes in which neighboring digits differ at most by 1.

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Haskell

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A252490 Numbers whose set of digits is simply connected, with 9 and 0 considered as neighbors.

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A032987 Numbers with the property that all pairs of consecutive base-10 digits differ by more than 2.

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Mathematica

A252481 Numbers whose set of digits is simply connected to 1.

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PARI

Python

A284046 Numbers k, not ending in 0, such that the consecutive digits of k^2 differ by 0 or 1.

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Mathematica

A357142 Nonnegative numbers all of whose pairs of consecutive decimal digits are adjacent digits, where 9 and 0 are considered adjacent.

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