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A381247 Positive integers m such that m * (m+1) contains at least 8 decimal digits that are in neither m nor m+1.

%I A381247 #25 Feb 19 2025 10:21:00
%S A381247 77776,88888,444554,554544,655555,656565,656665,888787,888888,1111121,
%T A381247 1212121,1222121,2212121,2222232,2222332,2223222,2232222,2322322,
%U A381247 2332222,3223232,3223332,3232222,3233322,3322332,3323232,3443443,4334443,4343443,4444443,4544444,5444444,5445554,5455454
%N A381247 Positive integers m such that  m * (m+1) contains at least 8 decimal digits that are in neither m nor m+1.
%C A381247 Theorem: The sequence contains all m(n) = A002282(2n+1) = 8*R(2n+1) with n >= 2, where R(n) = (10^n-1)/9 = A002275(n) is the n-th repunit. - Proof: for these m, whe have m*(m+1) == 65432 (mod 10^5) and floor(m*(m+1)/10^(4n-2)) = 7091, as one can check by direct calculation. Therefore all the digits 0 through 7 are present in these numbers. - Actually, as n grows, the string of initial digits consists of increasingly more copies of "790123456", and the final digits are increasingly many copies of "098765432". - _M. F. Hasler_, Feb 18 2025
%H A381247 Robert Israel, <a href="/A381247/b381247.txt">Table of n, a(n) for n = 1..10000</a>
%e A381247 77776*77777 = 6049183952 which has 8 new digits.
%e A381247 88888*88889 = 79011'65432, (with ' inserted to separate 1st and 2nd half of digits)
%e A381247 888888*888889 = 7901233'8765432,
%e A381247 8888888*8888889 = 790123456'098765432,
%e A381247 88888888*88888889 = 79012345678'32098765432,
%e A381247 888888888*888888889 = 7901234567900'5432098765432, etc.
%e A381247 It appears that only the last digit of the first part may be off by +-1 from the digit that would follow according to the periodic pattern. - _M. F. Hasler_, Feb 18 2025
%p A381247 filter:= proc(t) nops(convert(convert(t*(t+1),base,10),set) minus
%p A381247 (convert(convert(t,base,10),set) union convert(convert(t+1,base,10),set))) = 8 end proc [Maple code is missing some punctuation! - _N. J. A. Sloane_, Feb 19 2025]
%p A381247 f:= proc(d) local Cands,i,s,S;
%p A381247   Cands:= {seq(i*(10^d-1)/9, i=1..9), seq(seq(i*(10^d-1)/9 + add(10^s,s=S),i=1..8),S=combinat:-powerset({$1..d-1}))};
%p A381247   sort(convert(select(filter,Cands),list))
%p A381247 end proc:
%p A381247 seq(op(f(d)),d=1..7); # _Robert Israel_, Feb 18 2025
%o A381247 (PARI) select( {is_A381247(n)=#setminus(Set(digits(n*(n+1))), Set(concat(digits(n), digits(n+1))))>7}, [10^4..10^5])
%o A381247 for(n=1,1e7, is_A381247(n)&& print1(n", "))
%Y A381247 Cf. A381248.
%K A381247 nonn,base
%O A381247 1,1
%A A381247 _Ali Sada_ and _M. F. Hasler_, Feb 17 2025
%E A381247 More terms from _Michel Marcus_, Feb 18 2025