A061810 -id:A061810 - cp's OEIS frontend

A014263 Numbers that contain even digits only.

0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, 242, 244, 246, 248, 260, 262, 264, 266, 268, 280, 282, 284, 286, 288, 400, 402, 404, 406, 408, 420, 422, 424

Offset: 1

N. J. A. Sloane

The set of real numbers between 0 and 1 that contain no odd digits in their decimal expansion has Hausdorff dimension log 5 / log 10.
Integers written in base 5 and then doubled (in base 10). - Franklin T. Adams-Watters, Mar 15 2006
The carryless mod 10 "even" numbers (cf. A004529) sorted and duplicates removed. - N. J. A. Sloane, Aug 03 2010.
Complement of A007957; A196564(a(n)) = 0; A103181(a(n)) = 0. - Reinhard Zumkeller, Oct 04 2011
If n-1 is represented as a base-5 number (see A007091) according to n-1 = d(m)d(m-1)…d(3)d(2)d(1)d(0) then a(n)= Sum_{j=0..m} c(d(j))*10^j, where c(k)=0,2,4,6,8 for k=0..4. - Hieronymus Fischer, Jun 03 2012

			a(1000) = 24888.
a(10^4) = 60888.
a(10^5) = 22288888.
a(10^6) = 446888888.

K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 19.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.
Index entries for sequences related to carryless arithmetic

Subsequence of A059708.

Cf. A061810, A061811, A007091, A014261, A046034, A052382, A084544, A089581, A084984, A017042, A001743, A202267, A202268, A194182, A196563.

a014263 n = a014263_list !! (n-1)
a014263_list = filter (all (`elem` "02468") . show) [0,2..]
-- Reinhard Zumkeller, Jul 05 2011

[n: n in [0..424] | Set(Intseq(n)) subset [0..8 by 2]];  // Bruno Berselli, Jul 19 2011

a:= proc(m) local L,i;
  L:= convert(m-1,base,5);
  2*add(L[i]*10^(i-1),i=1..nops(L))
end proc:
seq(a(i),i=1..100); # Robert Israel, Apr 07 2016

Select[Range[450], And@@EvenQ[IntegerDigits[#]]&] (* Harvey P. Dale, Jan 30 2011 *)
FromDigits/@Tuples[{0,2,4,6,8},3] (* Harvey P. Dale, Jul 07 2025 *)

a(n) = 2*fromdigits(digits(n-1, 5), 10); \\ Michel Marcus, Nov 04 2022

is(n)=#setminus(Set(digits(n)), [0,2,4,6,8])==0 \\ Charles R Greathouse IV, Mar 03 2025

from sympy.ntheory.digits import digits
def a(n): return int(''.join(str(2*d) for d in digits(n, 5)[1:]))
print([a(n) for n in range(58)]) # Michael S. Branicky, Jan 13 2022

from itertools import count, islice, product
def agen(): # generator of terms
    yield 0
    for d in count(1):
        for first in "2468":
            for rest in product("02468", repeat=d-1):
                yield int(first + "".join(rest))
print(list(islice(agen(), 58))) # Michael S. Branicky, Jan 13 2022

A045888(a(n)) = 0. - Reinhard Zumkeller, Aug 25 2009
a(n) = A179082(n) for n <= 25. - Reinhard Zumkeller, Jun 28 2010
From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = ((2*b_m(n)) mod 8 + 2)*10^m + Sum_{j=0..m-1} ((2*b_j(n)) mod 10)*10^j, where n>1, b_j(n) = floor((n-1-5^m)/5^j), m = floor(log_5(n-1)).
a(1*5^n+1) = 2*10^n.
a(2*5^n+1) = 4*10^n.
a(3*5^n+1) = 6*10^n.
a(4*5^n+1) = 8*10^n.
a(n) = 2*10^log_5(n-1) for n=5^k+1,
a(n) < 2*10^log_5(n-1), else.
a(n) > (8/9)*10^log_5(n-1) n>1.
a(n) = 2*A007091(n-1), iff the digits of A007091(n-1) are 0 or 1.
G.f.: g(x) = (x/(1-x))*Sum_{j>=0} 10^j*x^5^j *(1-x^5^j)* (2+4x^5^j+ 6(x^2)^5^j+ 8(x^3)^5^j)/(1-x^5^(j+1)).
Also: g(x) = 2*(x/(1-x))*Sum_{j>=0} 10^j*x^5^j * (1-4x^(3*5^j)+3x^(4*5^j))/((1-x^5^j)(1-x^5^(j+1))).
Also: g(x) = 2*(x/(1-x))*(h_(5,1)(x) + h_(5,2)(x) + h_(5,3)(x) + h_(5,4)(x) - 4*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*(x^5^j)^k/(1-(x^5^j)^5). (End)
a(5*n+i-4) = 10*a(n) + 2*i for n >= 1, i=0..4. - Robert Israel, Apr 07 2016
Sum_{n>=2} 1/a(n) = A194182. - Bernard Schott, Jan 13 2022

Examples and crossrefs added by Hieronymus Fischer, Jun 06 2012

A350536 a(n) is the smallest proper multiple of 2n+1 which contains only odd digits, or -1 if no such multiple exists.

Original entry on oeis.org

3, 9, 15, 35, 99, 33, 39, 75, 51, 57, 315, 115, 75, 135, 319, 93, 99, 175, 111, 117, 533, 559, 135, 517, 539, 153, 159, 715, 171, 177, 793, 315, 195, 335, 759, 355, 511, 375, 539, 395, 1377, 913, 595, 957, 979, 1911, 1395, 1995, 3395, 9999, 1111, 515, 315, 535, 1199, 333

Offset: 0

Bernard Schott, Jan 04 2022

Generalization of the problem 1/2 of International Mathematical Talent Search, round 2 (see link and 2nd example).

If the escape clause is used, it will be necessarily for terms coming from n = 12 + 25*k, k >= 0.

			a(10) = 315 = 21 * 15 is the smallest multiple of 21 which contains only odd digits.
a(4998) = 33339995 = 9997 * 3335 is the smallest multiple of 9997 which contains only odd digits, so this is the answer to the IMTS problem.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
International Mathematical Talent Search, Problem 1/2, Round 2.
Index to sequences related to Olympiads and other Mathematical Competitions.

Cf. A061810, A061829, A078221, A296009, A350538.

Terms belong to A014261.

a[n_] := Module[{m = 2*n + 1, k}, k = 3*m; While[!AllTrue[IntegerDigits[k], OddQ], k += 2*m]; k]; Array[a, 50, 0] (* Amiram Eldar, Jan 04 2022 *)

isok(k) = my(d=digits(k)); #d == #select(x->((x%2)==1), d);
a(n) = my(k=6*n+3); while (!isok(k), k+=4*n+2); k; \\ Michel Marcus, Jan 04 2022

from itertools import product, count
def A350536(n):
    m = 2*n+1
    for l in count(len(str(m))):
        for s in product('13579',repeat=l):
            k = int(''.join(s))
            if k > m and k % m == 0:
                return k # Chai Wah Wu, Jan 11 2022

More terms from Michel Marcus, Jan 04 2022

A061811 Multiples of 3 with all even digits.

Original entry on oeis.org

0, 6, 24, 42, 48, 60, 66, 84, 204, 222, 228, 240, 246, 264, 282, 288, 402, 408, 420, 426, 444, 462, 468, 480, 486, 600, 606, 624, 642, 648, 660, 666, 684, 804, 822, 828, 840, 846, 864, 882, 888, 2004, 2022, 2028, 2040, 2046, 2064, 2082, 2088, 2202, 2208

Offset: 1

Amarnath Murthy, May 28 2001

The numbers b(d) of terms from 10^(d-1) to 10^d satisfy the recurrence b(d) = 6 b(d-1) - 6 b(d-2) + 5 b(d-3) with b(1)=1, b(2)=6, b(3)=33. For d >= 4, b(d) = (3*A276508(d) - 10*A276508(d-1) + 3*A276508(d-2))/7. - Robert Israel, Feb 15 2017

			228 has all even digits and 228 = 3*76.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A061810, A276508.

N:= 4: # for all terms < 10^N
E[1,0]:= {6}:
E[1,1]:= {4}:
E[1,2]:= {2,8}:
for n from 2 to N do
  for j from 0 to 2 do
    E[n,j]:= map(t -> (10*t, 10*t+6),E[n-1,j]) union
             map(t -> (10*t+2, 10*t+8), E[n-1,j+1 mod 3]) union
           map(t -> 10*t+4, E[n-1,j+2 mod 3]);
od od:
A:=sort([0,seq(op(E[i,0]),i=1..N)]); # Robert Israel, Feb 15 2017

Select[3*Range[0,800],AllTrue[IntegerDigits[#],EvenQ]&] (* Harvey P. Dale, May 03 2025 *)

is(n)=n%3==0 && #setintersect(Set(digits(n)), [1,3,5,7,9])==0 \\ Charles R Greathouse IV, Feb 15 2017

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

Offset corrected by Charles R Greathouse IV, Feb 15 2017

A228096 Numbers consisting of only odd digits such that no permutation of its digits yields a prime.

Original entry on oeis.org

1, 9, 15, 33, 39, 51, 55, 57, 75, 77, 93, 99, 111, 117, 135, 153, 155, 159, 171, 177, 195, 315, 333, 339, 351, 355, 357, 375, 393, 399, 513, 515, 519, 531, 535, 537, 551, 553, 555, 559, 573, 579, 591, 595, 597, 711, 717, 735, 753, 759, 771, 777, 795

Offset: 1

Jayanta Basu, Aug 10 2013

Apart from the first term, A061810 is a subsequence. Conjecture: a(n) ~ A061810(n). - Charles R Greathouse IV, Feb 15 2017

			51 is a member since it consists of only odd digits and both 15 and 51 are composites.

Jayanta Basu, Table of n, a(n) for n = 1..1000

Subsequence of A067013.

Cf. A067013, A061810.

Select[Range[800], And @@ OddQ[x = IntegerDigits[#]] && Count[FromDigits /@ Permutations[x], _?PrimeQ] == 0 &]
Table[FromDigits/@Select[Tuples[Range[1,9,2],n],NoneTrue[FromDigits/@ Permutations[#],PrimeQ]&],{n,3}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 09 2019 *)

A061817 Multiples of 9 containing only odd digits.

Original entry on oeis.org

9, 99, 117, 135, 153, 171, 315, 333, 351, 513, 531, 711, 999, 1179, 1197, 1359, 1377, 1395, 1539, 1557, 1575, 1593, 1719, 1737, 1755, 1773, 1791, 1917, 1935, 1953, 1971, 3159, 3177, 3195, 3339, 3357, 3375, 3393, 3519, 3537, 3555, 3573, 3591, 3717, 3735

Offset: 1

Amarnath Murthy, May 28 2001

			117 = 9*13 is a term.

Index entries for 10-automatic sequences.

Subsequence of A014261.

Cf. A061810, A014263, A061814, A061815, A061816.

Select[9Range[500],And@@OddQ[IntegerDigits[#]]&] (* Harvey P. Dale, May 30 2013 *)

is(n)=n%9==0 && #setintersect(Set(digits(n)), [0,2,4,6,8])==0 \\ Charles R Greathouse IV, Feb 15 2017

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 30 2001

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A014263 Numbers that contain even digits only.

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A350536 a(n) is the smallest proper multiple of 2n+1 which contains only odd digits, or -1 if no such multiple exists.

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A061811 Multiples of 3 with all even digits.

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A228096 Numbers consisting of only odd digits such that no permutation of its digits yields a prime.

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Mathematica

A061817 Multiples of 9 containing only odd digits.

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Programs

Mathematica

PARI

Extensions