A014263 -id:A014263 - cp's OEIS frontend

A136904 Numbers k such that k and k^2 use only the digits 0, 2, 4, 6 and 8.

0, 2, 8, 20, 22, 68, 80, 200, 202, 220, 262, 668, 680, 800, 2000, 2002, 2020, 2022, 2200, 2202, 2620, 6668, 6680, 6800, 8000, 8262, 20000, 20002, 20020, 20022, 20200, 20220, 20602, 22000, 22002, 22020, 24622, 26200, 66668, 66680, 66800, 68000, 80000, 82620, 200000, 200002, 200020, 200022, 200200, 200202, 200220, 202000, 202002, 202200, 206020

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

Subsequence of A014263. - Michel Marcus, Oct 04 2013

			828268460602^2 = 686028642828006826202404.

Jonathan Wellons, Table of n, a(n) for n = 1..1038
J. Wellons, Tables of Shared Digits [archived]

Cf. A014263.

Select[Range[0,206050],AllTrue[Join[IntegerDigits[#],IntegerDigits[#^2]],EvenQ] &] (* Stefano Spezia, Jul 03 2025 *)

evdigs(n) = {if (n==0, return (1)); digs = digits(n); for (i = 1, #digs, if (digs[i] % 2, return (0));); return (1);}
isok(n) = evdigs(n) && evdigs(n^2); \\ Michel Marcus, Oct 04 2013

A350538 a(n) is the smallest proper multiple of n which contains only even digits.

Original entry on oeis.org

2, 4, 6, 8, 20, 24, 28, 24, 288, 20, 22, 24, 26, 28, 60, 48, 68, 288, 228, 40, 42, 44, 46, 48, 200, 208, 486, 84, 406, 60, 62, 64, 66, 68, 280, 288, 222, 228, 468, 80, 82, 84, 86, 88, 2880, 460, 282, 240, 686, 200, 204, 208, 424, 486, 220, 224, 228, 406, 826

Offset: 1

Bernard Schott, Jan 05 2022

Inspired by the problem 1/2 of International Mathematical Talent Search, round 2 (see link).

Differs from A061807 when n is in A014263. - Michel Marcus, Jan 05 2022

			a(9) = 288 = 32 * 9 is the smallest multiple of 9 which contains only even digits.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
International Mathematical Talent Search, Problem 1/2, Round 2.

Cf. A061807, A350536.

Terms belong to A014263.

a[n_] := Module[{k = 2*n}, While[! AllTrue[IntegerDigits[k], EvenQ], k += n]; k]; Array[a, 60] (* Amiram Eldar, Jan 05 2022 *)

a(n) = my(k=2); while(#select(x->((x%2) == 1), digits(k*n)), k++); k*n; \\ Michel Marcus, Jan 12 2022

def a(n):
    m, inc = 2*n, n if n%2 == 0 else 2*n
    while not set(str(m)) <= set("02468"): m += inc
    return m
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Jan 05 2022

from itertools import count, product
def A350538(n):
    for l in count(len(str(n))-1):
        for a in '2468':
            for b in product('02468',repeat=l):
                k = int(a+''.join(b))
                if k > n and k % n == 0:
                    return k # Chai Wah Wu, Jan 12 2022

More terms from Michael S. Branicky, Jan 05 2022

A351893 Numbers that contain only even digits in their factorial-base representation.

Original entry on oeis.org

0, 4, 12, 16, 48, 52, 60, 64, 96, 100, 108, 112, 240, 244, 252, 256, 288, 292, 300, 304, 336, 340, 348, 352, 480, 484, 492, 496, 528, 532, 540, 544, 576, 580, 588, 592, 1440, 1444, 1452, 1456, 1488, 1492, 1500, 1504, 1536, 1540, 1548, 1552, 1680, 1684, 1692, 1696

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms are multiples of 4 (A008586).

			4 is a term since its factorial-base presentation, 20, has only even digits.
16 is a term since its factorial-base presentation, 220, has only even digits.

Amiram Eldar, Table of n, a(n) for n = 1..14400 (terms below 10!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A008586, A351894.
Subsequence: A052849 \ {2}.
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[0, max!, 2], AllTrue[fctBaseDigits[#], EvenQ] &]

A007957 Numbers that contain an odd digit.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 85, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

R. Muller

Complement of A014263; A196564(a(n)) > 0; A103181(a(n)) > 0. - Reinhard Zumkeller, Oct 04 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Index entries for 10-automatic sequences.

Cf. A014263, A103181, A196564.

import Data.List (findIndices)
a007957 n = a007957_list !! (n-1)
a007957_list = findIndices (> 0) a196564_list
a196564 n = length [d | d <- show n, d `elem` "13579"]
a196564_list = map a196564 [0..]
-- Reinhard Zumkeller, Oct 04 2011

Select[Range[100],Count[IntegerDigits[#],?OddQ]>0&] (* _Harvey P. Dale, Sep 06 2017 *)

is(n)=my(v=vecsort(eval(Vec(Str(n)))%2,,8));v[#v] \\ Charles R Greathouse IV, Jul 25 2012

a(n) = n + O(n^0.69897...) where the constant is A153268. - Charles R Greathouse IV, Jul 25 2012

A103181 In decimal representation of n: replace all even digits with 0 and all odd digits with 1.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 0, 1, 0, 1, 0, 1

Offset: 0

Reinhard Zumkeller, Mar 18 2005

			199->'111': a(199)=111; 200->'000': a(200)=0;
299->'011': a(299)=11; 300->'100': a(300)=100.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A000035, A014263.

import Data.List (unfoldr); import Data.Tuple (swap)
a103181_list = map a103181 [0..]
a103181 n = foldl f 0 $ reverse $ unfoldr g n where
   f v d = 10 * v + mod d 2
   g x = if x == 0 then Nothing else Just $ swap $ divMod x 10
-- Reinhard Zumkeller, Oct 04 2011

Table[FromDigits[If[EvenQ[#],0,1]&/@IntegerDigits[n]],{n,0,90}] (* Harvey P. Dale, Nov 08 2022 *)

a(n)=subst(Pol(apply(k->k%2, digits(n))),'x,10) \\ Charles R Greathouse IV, Jul 16 2013

def A103181(n): return int(''.join(str(int(d) % 2) for d in str(n))) # Chai Wah Wu, Apr 09 2022

n = Sum(d(k)*10^k: 0<=d(k)<10) -> a(n) = Sum((d(k) mod 2)*10^k).

a(A014263(n)) = 0. - Reinhard Zumkeller, Oct 04 2011

A168501 Numbers without the decimal digits 2, 4 and 6.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 30, 31, 33, 35, 37, 38, 39, 50, 51, 53, 55, 57, 58, 59, 70, 71, 73, 75, 77, 78, 79, 80, 81, 83, 85, 87, 88, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 130, 131, 133

Offset: 1

Juri-Stepan Gerasimov, Nov 27 2009

Numbers that contain non-single or nonisolated digits only (see the first seven values of A167707 for the digits that are not 2, 4, or 6).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A014263, A167707.

[n: n in [0..200] | IsEmpty(Set([2,4,6]) meet Set(Intseq(n)))]; // Bruno Berselli, Jul 25 2016

Select[Select[Select[Range[0, 4500], FreeQ[IntegerDigits@#, 2] &], FreeQ[IntegerDigits@#, 4] &], FreeQ[IntegerDigits@#, 6] &] (* G. C. Greubel, Jul 24 2016 *)

Corrected (114 removed) by R. J. Mathar, Jun 04 2010

A342846 Number of distinct odd numbers visible as proper substrings of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2

Offset: 1

Sean A. Irvine, Mar 24 2021

Here substrings are contiguous.

a(A164766(n)) = n and a(m) <> n for m < A164766(n); a(A014263(n)) = 0. - Reinhard Zumkeller, Aug 25 2009

			a(10)=1 since we can see 1 as a proper substring of 10.
a(105)=2 since we can see 1, 5.
a(132)=3 because we can see 1, 3, 13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..20000
Sean A. Irvine, Java program (github)

Cf. A045887, A045888, A342845.

import Data.List (isInfixOf)
a045888 n = length $ filter (`isInfixOf` (show n)) $ map show [1, 3..n-1]
-- Reinhard Zumkeller, Jul 19 2011

def a(n):
  s, eset = str(n), set()
  for i in range(len(s)):
    for j in range(i+1, len(s)+1):
      if s[j-1] in "13579" and j-i < len(s): # odd and proper substring
        eset.add(int(s[i:j]))
  return len(eset)
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Mar 24 2021

A067044 Smallest positive k such that k*n contains only even digits.

Original entry on oeis.org

2, 1, 2, 1, 4, 1, 4, 1, 32, 2, 2, 2, 2, 2, 4, 3, 4, 16, 12, 1, 2, 1, 2, 1, 8, 1, 18, 1, 14, 2, 2, 2, 2, 2, 8, 8, 6, 6, 12, 1, 2, 1, 2, 1, 64, 1, 6, 1, 14, 4, 4, 4, 8, 9, 4, 4, 4, 7, 14, 1, 4, 1, 14, 1, 4, 1, 4, 1, 12, 4, 4, 4, 28, 3, 8, 3, 6, 6, 34, 1, 6, 1, 8, 1, 8, 1, 24, 1, 32, 32, 22, 5, 22, 3

Offset: 1

Amarnath Murthy, Dec 29 2001

No multiple of 10 can appear in this sequence. - M. F. Hasler, Mar 07 2025

			a(7) = 4 as among the multiples of 7 (i.e., 7, 14, 21, 28...), 28 is the smallest multiple with only even digits and a(7)= 28/7 = 4.
a(16) = 3 is the first odd term > 1, a(n = 54, 58, 74, 76, 92, 94, 96, 98, ...) are the next examples, cf. A380874. - _M. F. Hasler_, Mar 03 2025

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A061807, A380874.

Cf. A014263 (numbers with only even digits), A007091 (numbers in base 5).

Table[k = n; While[Length[Intersection[{1, 3, 5, 7, 9}, IntegerDigits[k]]] > 0, k = k + n]; k/n, {n, 100}] (* T. D. Noe, Jun 03 2013 *)
sk[n_]:=Module[{k=1},While[!AllTrue[IntegerDigits[k*n],EvenQ],k++];k]; Array[sk,100] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 27 2015 *)

apply( {A067044(n, f=1+n%2)=forstep(a=f*n, oo, f*n, digits(a)%2||return(a/n))}, [1..99]) \\ M. F. Hasler, Mar 03 2025

A067044 = lambda n: next(k for k in range(1+n%2, 9<<99, 1+n%2)if not any(int(d)&1 for d in str(n*k))) # M. F. Hasler, Mar 03 2025

From M. F. Hasler, Mar 07 2025: (Start)
There is an explicit formula for many values of n:
a(n) = 1 if n has only even digits <=> n is in A014263, else:
a(n) = 2 if n has only digits < 5 <=> n is in A007091;
a(m*(10^k-1)) = 8*round(10^k/6)^2/m for m = 1, 2, 4 or 8 and any k > 0;
a(5*(10^k-1)) = 16*round(10^k/6)^2 for any k > 0;
a(50*m + {5 or 15}) = 4 if m has all digits < 5. (End)

More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), May 06 2002

Data corrected by Paul Tek, Jun 03 2013

A071241 Arithmetic mean of k and R(k) where k is the n-th nonnegative number using only even digits and R(k) is its digit reversal (A004086).

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 22, 33, 44, 55, 22, 33, 44, 55, 66, 33, 44, 55, 66, 77, 44, 55, 66, 77, 88, 101, 202, 303, 404, 505, 121, 222, 323, 424, 525, 141, 242, 343, 444, 545, 161, 262, 363, 464, 565, 181, 282, 383, 484, 585, 202, 303, 404, 505, 606, 222, 323, 424, 525

Offset: 0

Amarnath Murthy, May 20 2002

Conjecture: 101 is the largest prime term, the only other primes being 2 and 11.

The conjecture is false: for example, 181 and 383 are prime terms. There are 150 prime terms less than 75000. - Harvey P. Dale, Sep 02 2016

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Cf. A004086, A014263, A071240, A071242.

reversal := proc(n) local i, len, new, temp: new := 0: temp := n: len := floor(log[10](n+.1))+1: for i from 1 to len do new := new+irem(temp, 10)*10^(len-i): temp := floor(temp/10): od: RETURN(new): end: alleven := proc(n) local i, flag, len, temp: temp := n: flag := 1: if n=0 then flag := 0 fi: len := floor(log[10](n+.1))+1: for i from 1 to len do if irem(temp, 10) mod 2 = 0 then temp := floor(temp/10) else flag := 0 fi: od: RETURN(flag): end: for n from 0 to 500 by 2 do if alleven(n) = 1 then printf(`%d,`,(n+reversal(n))/2) fi: od: # James Sellers, May 28 2002

Mean[{#,IntegerReverse[#]}]&/@(FromDigits/@Tuples[{0,2,4,6,8},3]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 02 2016 *)

{k + R(k)}/2 where k uses only odd digits 0, 2, 4, 6 and 8.

a(n) = (A014263(n) + A004086(A014263(n))) / 2. - Sean A. Irvine, Jul 06 2024

More terms from James Sellers, May 28 2002

Corrected by Harvey P. Dale, Sep 02 2016

A117978 Triangular numbers with only even digits.

Original entry on oeis.org

0, 6, 28, 66, 406, 666, 820, 2080, 2628, 8646, 28680, 42486, 48828, 64620, 66066, 80200, 84666, 200028, 204480, 228826, 264628, 288420, 426426, 446040, 468028, 484620, 600060, 626640, 644680, 686206, 828828, 886446, 2222886, 2248260, 2862028, 2888406

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 03 2006

There are infinitely many terms. For example all 88...88 6 44...44 6 with an equal number of 8's and 4's (their triangular indices being A185127). - Shyam Sunder Gupta, Aug 16 2025

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 82-125.

Intersection of A000217 and A014263.

Cf. A117960, A185127.

Select[Accumulate[Range[0,5000]],Count[IntegerDigits[#],?(OddQ)] ==0&] (* _Harvey P. Dale, Oct 21 2011 *)

Corrected (a(32) was in error) and extended by Harvey P. Dale, Oct 21 2011

cp's OEIS Frontend

A136904 Numbers k such that k and k^2 use only the digits 0, 2, 4, 6 and 8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A350538 a(n) is the smallest proper multiple of n which contains only even digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Extensions

A351893 Numbers that contain only even digits in their factorial-base representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A007957 Numbers that contain an odd digit.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A103181 In decimal representation of n: replace all even digits with 0 and all odd digits with 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A168501 Numbers without the decimal digits 2, 4 and 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A342846 Number of distinct odd numbers visible as proper substrings of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs