A074157 -id:A074157 - cp's OEIS frontend

A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.

2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 2, 4, 4, 3, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 3, 2, 2, 0, 2, 4, 2, 2, 2, 2, 4, 2, 12, 0, 4, 2, 2, 2, 2, 2, 2, 2, 2, 0, 4, 5, 3, 2, 2, 2, 2, 3, 2, 0, 4, 2, 2, 3, 4, 2, 2, 2, 2, 0, 3, 2, 2, 3

Offset: 1

Gonzalo Martínez, Dec 09 2024

It is observed that a(10*m) = 0 and a(100*m + 5) = 0 for all positive integers m.

If m is a repdigit number (A010785) that does not have the digit 9, then a(m) = 2 and if m = 99...9, with t 9's, then a(m) = 11...2, i.e., (t - 1) 1's followed by 2, since 99...9 * 11...2 equals (t - 1) 1's followed by (t - 1) 8's, where k = 11...2 is the smallest number with this property. In other words, a(A002283(m)) = A047855(m), for all positive integers m.

			a(12) = 3, since 12*1 = 12, 12*2 = 24 have digits in common with 12, while 12*3 = 36 does not.

Dominic McCarty, Table of n, a(n) for n = 1..1000 (synthesized from _Michel Marcus_'s data at A074157)

Cf. A074157, A010785, A002283, A047855.

a(n) = A074157(n)/n.

A076924 Smallest multiple of the n-th prime not containing any of its digits, or 0 if no such number exists.

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 111, 82, 86, 188, 106, 118, 244, 134, 284, 146, 158, 166, 267, 388, 2222, 824, 428, 327, 226, 508, 262, 548, 278, 2086, 302, 628, 489, 334, 692, 358, 362, 382, 772, 2364, 2388, 633, 446, 454, 458, 466, 478, 3856, 4769

Offset: 1

Amarnath Murthy, Oct 17 2002

a(n) = 0 if and only if prime(n) contains all decimal digits with the possible exception of 0. Otherwise, if n > 3 and p=prime(n) does not contain the nonzero digit k, the repdigit (10^m-1)*k/9 is a multiple of p where m is the order of 10 mod p. - Robert Israel, Jul 21 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A074157, A076925.

f:= proc(n) local p, k,r,D;
  D:= convert(convert(n,base,10),set);
  for k from 2 to 10^(1+ilog10(n)) do
    r:= k*n;
    if convert(convert(r,base,10),set) intersect D = {} then return r fi
  od;
  error ("search failed for n=%1",n)
end proc:
seq(f(ithprime(i)),i=1..100); # Robert Israel, Jul 21 2020

smp[p_]:=Module[{k=2},While[ContainsAny[IntegerDigits[k*p], IntegerDigits[ p]],k++];k*p]; Table[smp[p],{p,Prime[Range[60]]}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 19 2019 *)

a(n) = A074157(prime(n)). - Robert Israel, Jul 21 2020

Corrected and extended by Ray Chandler, Feb 11 2005

A379673 Records in A378865.

Original entry on oeis.org

2, 3, 4, 12, 22, 33, 38, 65, 112, 127, 222, 333, 625, 1112, 1247, 2222, 3333, 3612, 546875, 1640625, 48828125, 452734375, 4150390625, 225830078125, 11444091796875, 52674606084709, 155879785509415, 407757583839192, 342846312788889239, 434661425189366827, 483009317939766618

Offset: 1

Gonzalo Martínez, Jan 02 2025

nonn

Cf. A074157, A378865, A385290.

a(19) onwards corrected by Jinyuan Wang and Sean A. Irvine, Jun 19 2025

More terms from Jinyuan Wang, Jun 25 2025

cp's OEIS Frontend

A378865 a(n) is the smallest positive integer k such that n*k uses none of the digits of n, or 0 if no such k exists.

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A076924 Smallest multiple of the n-th prime not containing any of its digits, or 0 if no such number exists.

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A379673 Records in A378865.

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