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

%I A380878 #14 Feb 07 2025 16:00:50
%S A380878 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,
%T A380878 56,57,58,65,66,76,77,78,83,85,92,143,144,148,154,156,165,175,188,194,
%U A380878 195,222,232,237,242,257,265,292,294,303,307,312,313,322,332,333,334,343,344,375,377,387,392
%N A380878 Numbers k such that k*(k+1) shares no decimal digits with k or k+1.
%D A380878 If k is a term, then at least one of k and k+1 is in A375211.
%D A380878 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.
%H A380878 Robert Israel, <a href="/A380878/b380878.txt">Table of n, a(n) for n = 1..5000</a>
%e A380878 a(8) = 15 is a term because 15 * 16 = 240 and none of its digits 2, 4, 0 occur in 15 or 16.
%e A380878 16 is not a term because 16 * 17 = 272 contains the digit 7 which also occurs in 17.
%p A380878 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) = {}:
%p A380878 select(filter, [$1..1000]);
%t A380878 A380878Q[k_] := Intersection[Join[IntegerDigits[k], IntegerDigits[k+1]], IntegerDigits[k*(k+1)]] == {};
%t A380878 Select[Range[500], A380878Q] (* _Paolo Xausa_, Feb 07 2025 *)
%o A380878 (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
%Y A380878 Cf. A375211.
%K A380878 nonn,base
%O A380878 1,1
%A A380878 _Robert Israel_, Feb 07 2025