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A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.

%I A380974 #44 Feb 21 2025 13:07:40
%S A380974 4,5,6,7,8,9,10,11,17,24,25,32,34,67,75,78,100,101,142,167,334,667,
%T A380974 1000,1001,1667,3334,6667,10000,10001,16667,33334,66667,100000,100001,
%U A380974 166667,333334,666667,1000000,1000001,1666667,3333334,6666667,10000000,10000001,16666667,33333334,66666667,100000000
%N A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.
%C A380974 Numbers k such that A002378(k-1) is in A031955.
%C A380974 Conjecture: all terms >= 334 are of the form 10...0, 10...01, 16...67, 3...34, or 6...67.
%C A380974 Last decimal digit of a(n)*(a(n)-1) is either 0, 2 or 6. - _Chai Wah Wu_, Feb 19 2025
%H A380974 Michael S. Branicky, <a href="/A380974/b380974.txt">Table of n, a(n) for n = 1..117</a>
%F A380974 Conjectured: for k >= 0,
%F A380974   a(20 + 5*k) = (10^(3+k) + 2)/6,
%F A380974   a(21 + 5*k) = (10^(3+k) + 2)/3,
%F A380974   a(22 + 5*k) = (2*10^(3+k)+1)/3,
%F A380974   a(23 + 5*k) = 10^(3+k),
%F A380974   a(24 + 5*k) = 10^(3+k) + 1.
%F A380974 Conjectured G.f.: (4*x + 5*x^2 + 6*x^3 + 7*x^4 + 8*x^5 - 35*x^6 - 45*x^7 - 55*x^8 - 60*x^9 - 64*x^10 - 34*x^11 - 28*x^12 - 27*x^13 - 50*x^14 - 109*x^15 - 107*x^16 - 152*x^17 - 163*x^18 - 425*x^19 - 418*x^20 - 274*x^21 - 113*x^22 + 229*x^23 + 109*x^24 + 580*x^25 + 440*x^26 + 330*x^27 + 10*x^28 + 410*x^29)/(1 - 11 * x^5 + 10 * x^10).
%e A380974 a(10) = 23 is a term because 23 * 24 = 552 contains two different digits 2 and 5.
%p A380974 select(k -> nops(convert(convert(k*(k+1),base,10),set)) = 2, [$1..10^6]);
%t A380974 Select[Range[10^7],Length[Union[IntegerDigits[#*(#-1)]]]==2&] (* _James C. McMahon_, Feb 13 2025 *)
%o A380974 (PARI) isok(k) = #Set(digits(k*(k-1))) == 2; \\ _Michel Marcus_, Feb 11 2025
%o A380974 (Python)
%o A380974 from math import isqrt
%o A380974 from itertools import count, combinations, product, islice
%o A380974 def A380974_gen(): # generator of terms
%o A380974     for n in count(1):
%o A380974         c = []
%o A380974         for a in combinations('0123456789',2):
%o A380974             if '0' in a or '2' in a or '6' in a:
%o A380974                 for b in product(a,repeat=n):
%o A380974                     if b[0] != '0' and b[-1] in {'0','2','6'} and b != (a[0],)*n and b != (a[1],)*n:
%o A380974                         m = int(''.join(b))
%o A380974                         q = isqrt(m)
%o A380974                         if q*(q+1)==m:
%o A380974                             c.append(q+1)
%o A380974         yield from sorted(c)
%o A380974 A380974_list = list(islice(A380974_gen(),30)) # _Chai Wah Wu_, Feb 19 2025
%Y A380974 Cf. A002378, A031955, A016069, A380984.
%K A380974 nonn,base
%O A380974 1,1
%A A380974 _Robert Israel_, Feb 11 2025