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

77776, 88888, 444554, 554544, 655555, 656565, 656665, 888787, 888888, 1111121, 1212121, 1222121, 2212121, 2222232, 2222332, 2223222, 2232222, 2322322, 2332222, 3223232, 3223332, 3232222, 3233322, 3322332, 3323232, 3443443, 4334443, 4343443, 4444443, 4544444, 5444444, 5445554, 5455454

Offset: 1

Author

Ali Sada and M. F. Hasler, Feb 17 2025

Keywords

• nonn
• base

Comments

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

Examples

			77776*77777 = 6049183952 which has 8 new digits.
88888*88889 = 79011'65432, (with ' inserted to separate 1st and 2nd half of digits)
888888*888889 = 7901233'8765432,
8888888*8888889 = 790123456'098765432,
88888888*88888889 = 79012345678'32098765432,
888888888*888888889 = 7901234567900'5432098765432, etc.
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
		

Links

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

Crossrefs

Cf. A381248.

Programs

Maple

filter:= proc(t) nops(convert(convert(t*(t+1),base,10),set) minus
(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]
f:= proc(d) local Cands,i,s,S;
  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}))};
  sort(convert(select(filter,Cands),list))
end proc:
seq(op(f(d)),d=1..7); # Robert Israel, Feb 18 2025

PARI

select( {is_A381247(n)=#setminus(Set(digits(n*(n+1))), Set(concat(digits(n), digits(n+1))))>7}, [10^4..10^5])
for(n=1,1e7, is_A381247(n)&& print1(n", "))

Extensions

More terms from Michel Marcus, Feb 18 2025