A350536 -id:A350536 - cp's OEIS frontend

A350537 Smallest number m > 1 such that (2n+1)*m = A350536(n) contains only odd digits.

3, 3, 3, 5, 11, 3, 3, 5, 3, 3, 15, 5, 3, 5, 11, 3, 3, 5, 3, 3, 13, 13, 3, 11, 11, 3, 3, 13, 3, 3, 13, 5, 3, 5, 11, 5, 7, 5, 7, 5, 17, 11, 7, 11, 11, 21, 15, 21, 35, 101, 11, 5, 3, 5, 11, 3, 3, 5, 3, 3, 11, 11, 3, 11, 15, 3, 3, 13, 7, 7, 11, 5, 11, 5, 13, 5, 9, 5, 47, 5

Offset: 0

Bernard Schott, Jan 12 2022

Record values of a(n) are 3, 5, 11, 15, 17, 21, 35, 101, 155, ...

			The smallest proper multiple of 21 = 2*10+1 with only odd digits is A350536(10) = 315, as 315 = 21 * 15, a(10) = 15.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A078221, A296009, A350536, A350538, A350697.

Table[m=2;While[Or@@EvenQ[IntegerDigits[(2n+1)*++m]]];m,{n,0,79}] (* Giorgos Kalogeropoulos, Jan 12 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/(2*n+1); \\ Michel Marcus, Jan 12 2022

a(n) = A350536(n) / (2n+1).

More terms from Michel Marcus, Jan 12 2022

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

A350697 Smallest number m > 1 such that n * m = A350538(n) contains only even digits.

Original entry on oeis.org

2, 2, 2, 2, 4, 4, 4, 3, 32, 2, 2, 2, 2, 2, 4, 3, 4, 16, 12, 2, 2, 2, 2, 2, 8, 8, 18, 3, 14, 2, 2, 2, 2, 2, 8, 8, 6, 6, 12, 2, 2, 2, 2, 2, 64, 10, 6, 5, 14, 4, 4, 4, 8, 9, 4, 4, 4, 7, 14, 4, 4, 4, 14, 7, 4, 4, 4, 3, 12, 4, 4, 4, 28, 3, 8, 3, 6, 6, 34, 3, 6, 3, 8, 5, 8

Offset: 1

Bernard Schott, Jan 12 2022

The smallest odd term is a(48) = 5 because 48*5 = 240.

Record values of a(n) are 2, 4, 32, 64, ...

			The smallest proper multiple of 9 with only even digits is A350538(9) = 288, as 288 = 9 * 32, a(9) = 32.

Cf. A061807, A350536, A350537, A350538.

a[n_] := Module[{k = 2*n}, While[! AllTrue[IntegerDigits[k], EvenQ], k += n]; k/n]; Array[a, 100] (* Amiram Eldar, Jan 12 2022 *)

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

a(n) = A350538(n) / n.

More terms from Michel Marcus, Jan 12 2022

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A350537 Smallest number m > 1 such that (2n+1)*m = A350536(n) contains only odd digits.

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

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A350697 Smallest number m > 1 such that n * m = A350538(n) contains only even digits.

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