A061829 -id:A061829 - cp's OEIS frontend

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

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

A061831 Multiples of 9 having only even digits.

Original entry on oeis.org

0, 288, 468, 486, 648, 666, 684, 828, 846, 864, 882, 2088, 2268, 2286, 2448, 2466, 2484, 2628, 2646, 2664, 2682, 2808, 2826, 2844, 2862, 2880, 4068, 4086, 4248, 4266, 4284, 4428, 4446, 4464, 4482, 4608, 4626, 4644, 4662, 4680, 4806, 4824, 4842, 4860, 6048, 6066

Offset: 1

Amarnath Murthy, May 29 2001

			648 = 9*72 is a term having all even digits.

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

Cf. A061829, A305826.

f:= proc(k) local L,t;
  L:= convert(k,base,5);
  t:= 2*add(L[i]*10^(i-1),i=1..nops(L));
  if t mod 9 = 0 then t fi
end proc:
map(f, [$0..1000]); # Robert Israel, Jun 10 2018

Select[9*Range[0, 700], AllTrue[IntegerDigits[#], EvenQ] &] (* Harvey P. Dale, Aug 07 2021 *)

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

For n >= 2, a(A305826(n)+1) = 2*10^n+88. - Robert Israel, Jun 10 2018

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 30 2001
Offset corrected by Charles R Greathouse IV, Feb 15 2017
Missing term 2880 inserted by Robert Israel, Jun 10 2018

A061830 Multiples of 5 having only even digits.

Original entry on oeis.org

0, 20, 40, 60, 80, 200, 220, 240, 260, 280, 400, 420, 440, 460, 480, 600, 620, 640, 660, 680, 800, 820, 840, 860, 880, 2000, 2020, 2040, 2060, 2080, 2200, 2220, 2240, 2260, 2280, 2400, 2420, 2440, 2460, 2480, 2600, 2620, 2640, 2660, 2680, 2800, 2820, 2840

Offset: 0

Amarnath Murthy, May 29 2001

			220 = 5*44 is a term having all even digits.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for 10-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A061829.

Select[5*Range[0,2000],And@@EvenQ[IntegerDigits[#]]&] (* or *) LinearRecurrence[ {1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{0,20,40,60,80,200,220,240,260,280,400,420,440,460,480,600,620,640,660,680,800,820,840,860,880,2000},50] (* Harvey P. Dale, Feb 24 2014 *)

is(n)=n%10==0 && #setintersect(Set(digits(n)), [1,3,5,7,9])==0 \\ Charles R Greathouse IV, Feb 15 2017

a(0)=0, a(1)=20, a(2)=40, a(3)=60, a(4)=80, a(5)=200, a(6)=220, a(7)=240, a(8)=260, a(9)=280, a(10)=400, a(11)=420, a(12)=440, a(13)=460, a(14)=480, a(15)=600, a(16)=620, a(17)=640, a(18)=660, a(19)=680, a(20)=800, a(21)=820, a(22)=840, a(23)=860, a(24)=880, a(25)=2000, a(n)=a(n-1)+ a(n-25)- a(n-26). - Harvey P. Dale, Feb 24 2014

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

A061832 Multiples of 11 having only even digits.

Original entry on oeis.org

0, 22, 44, 66, 88, 220, 242, 264, 286, 440, 462, 484, 660, 682, 880, 2002, 2024, 2046, 2068, 2200, 2222, 2244, 2266, 2288, 2420, 2442, 2464, 2486, 2640, 2662, 2684, 2860, 2882, 4004, 4026, 4048, 4202, 4224, 4246, 4268, 4400, 4422, 4444, 4466, 4488, 4620

Offset: 1

Amarnath Murthy, May 29 2001

			264 = 11*24 is a term having all even digits.

Index entries for 10-automatic sequences.

Cf. A061829, A061830, A061831.

is(n)=n%22==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

A061833 Multiples of 11 having only odd digits.

Original entry on oeis.org

11, 33, 55, 77, 99, 319, 517, 539, 715, 737, 759, 913, 935, 957, 979, 1111, 1133, 1155, 1177, 1199, 1331, 1353, 1375, 1397, 1551, 1573, 1595, 1771, 1793, 1991, 3113, 3135, 3157, 3179, 3311, 3333, 3355, 3377, 3399, 3531, 3553, 3575, 3597, 3751, 3773, 3795

Offset: 1

Amarnath Murthy, May 29 2001

			1353 = 11*123 is a term having all odd digits.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A061829-A061832.

Select[11*Range[400],AllTrue[IntegerDigits[#],OddQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 30 2015 *)

is(n)=n%22==11 && #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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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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A061831 Multiples of 9 having only even digits.

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A061830 Multiples of 5 having only even digits.

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A061832 Multiples of 11 having only even digits.

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A061833 Multiples of 11 having only odd digits.

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