A380974 Numbers k such that k*(k-1) is composed of exactly two different decimal digits.
4, 5, 6, 7, 8, 9, 10, 11, 17, 24, 25, 32, 34, 67, 75, 78, 100, 101, 142, 167, 334, 667, 1000, 1001, 1667, 3334, 6667, 10000, 10001, 16667, 33334, 66667, 100000, 100001, 166667, 333334, 666667, 1000000, 1000001, 1666667, 3333334, 6666667, 10000000, 10000001, 16666667, 33333334, 66666667, 100000000
Offset: 1
Examples
a(10) = 23 is a term because 23 * 24 = 552 contains two different digits 2 and 5.
Links
- Michael S. Branicky, Table of n, a(n) for n = 1..117
Programs
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Maple
select(k -> nops(convert(convert(k*(k+1),base,10),set)) = 2, [$1..10^6]);
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Mathematica
Select[Range[10^7],Length[Union[IntegerDigits[#*(#-1)]]]==2&] (* James C. McMahon, Feb 13 2025 *)
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PARI
isok(k) = #Set(digits(k*(k-1))) == 2; \\ Michel Marcus, Feb 11 2025
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Python
from math import isqrt from itertools import count, combinations, product, islice def A380974_gen(): # generator of terms for n in count(1): c = [] for a in combinations('0123456789',2): if '0' in a or '2' in a or '6' in a: for b in product(a,repeat=n): if b[0] != '0' and b[-1] in {'0','2','6'} and b != (a[0],)*n and b != (a[1],)*n: m = int(''.join(b)) q = isqrt(m) if q*(q+1)==m: c.append(q+1) yield from sorted(c) A380974_list = list(islice(A380974_gen(),30)) # Chai Wah Wu, Feb 19 2025
Formula
Conjectured: for k >= 0,
a(20 + 5*k) = (10^(3+k) + 2)/6,
a(21 + 5*k) = (10^(3+k) + 2)/3,
a(22 + 5*k) = (2*10^(3+k)+1)/3,
a(23 + 5*k) = 10^(3+k),
a(24 + 5*k) = 10^(3+k) + 1.
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).
Comments