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

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

1,1

References

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.

Links

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

Example

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.

Maple

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]);

Mathematica

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

Prog

(PARI) 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

Crossrefs

Cf. A375211.

Sequence in context: A245802 A085904 A032997 * A319724 A097546 A004837

Adjacent sequences: A380863 A380864 A380865 * A380879 A380880 A380881

Keyword

nonn,base,new

Author

Robert Israel, Feb 07 2025

Status

approved