A375211 -id:A375211 - cp's OEIS frontend

A129459 Slowest increasing sequence that starts with 0 and has property that multiplying two consecutive terms gives a number which shares at least one digit with at least one of the terms.

0, 1, 2, 6, 8, 10, 11, 12, 13, 14, 15, 17, 20, 21, 22, 24, 26, 27, 28, 29, 30, 31, 32, 35, 36, 37, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 53, 55, 57, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97

Offset: 0

Eric Angelini, May 29 2007

Terms computed by Stefan Steinerberger.

Includes all numbers that end with 0 or 1. - Robert Israel, Feb 06 2025

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A375211.

f:= proc(n) local Ln, Lk,k;
  Ln:= convert(convert(n,base,10),set);
  for k from n+1 do
     Lk:= convert(convert(k,base,10),set) union Ln;
     if convert(convert(n*k,base,10),set) intersect Lk <> {} then return k fi
  od
end proc:
Res:= 0: x:= 0:
for i from 1 to 100 do x:= f(x); Res:= Res,x od:
Res; # Robert Israel, Feb 06 2025

a = {0}; For[n = 1, n <= 100, n++, If[Length[Intersection[IntegerDigits[n*a[[ -1]]], Union[IntegerDigits[n], IntegerDigits[a[[ -1]]]]]] != 0, AppendTo[a, n]]]; a

A380878 Numbers k such that k*(k+1) shares no decimal digits with k or k+1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 15, 17, 18, 22, 24, 32, 33, 34, 37, 42, 43, 44, 45, 47, 48, 53, 54, 55, 56, 57, 58, 65, 66, 76, 77, 78, 83, 85, 92, 143, 144, 148, 154, 156, 165, 175, 188, 194, 195, 222, 232, 237, 242, 257, 265, 292, 294, 303, 307, 312, 313, 322, 332, 333, 334, 343, 344, 375, 377, 387, 392

Offset: 1

Robert Israel, Feb 07 2025

			a(8) = 15 is a term because 15 * 16 = 240 and none of its digits 2, 4, 0 occur in 15 or 16.
16 is not a term because 16 * 17 = 272 contains the digit 7 which also occurs in 17.

If k is a term, then at least one of k and k+1 is in A375211.
Contains (10^k - 4)/3, (10^k - 1)/3, (10^k + 2)/3, (2*10^k - 5)/3 and (2*10^k - 2)/3 for each k >= 1.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A375211.

filter:= t -> (convert(convert(t,base,10),set) union convert(convert(t+1,base,10),set)) intersect convert(convert(t*(t+1),base,10),set) = {}:
select(filter, [$1..1000]);

A380878Q[k_] := Intersection[Join[IntegerDigits[k], IntegerDigits[k+1]], IntegerDigits[k*(k+1)]] == {};
Select[Range[500], A380878Q] (* Paolo Xausa, Feb 07 2025 *)

isok(k) = my(s=Set(digits(k)), t=Set(digits(k+1)), u=Set(digits(k*(k+1)))); (#setintersect(s, u)==0) && (#setintersect(t, u)==0); \\ Michel Marcus, Feb 07 2025

cp's OEIS Frontend

A129459 Slowest increasing sequence that starts with 0 and has property that multiplying two consecutive terms gives a number which shares at least one digit with at least one of the terms.

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