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URL: https://oeis.org/A378048

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A378048
Numbers k such that k and k^2 together use at most 4 distinct decimal digits.
1
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35, 38, 40, 41, 45, 46, 50, 55, 56, 60, 63, 64, 65, 66, 68, 70, 74, 75, 76, 77, 80, 81, 83, 85, 88, 90, 91, 95, 96, 97, 99, 100, 101, 102, 105, 109, 110
OFFSET
1,3
COMMENTS
Problem: Is there a real constant c such that a(n) < n^c for all positive integers n?
All of A136808, A136809, A136816, ..., A137079 are subsequences. Many if not most terms of A058411, A058413, ... ("tridigital solutions") are also in this sequence; see also Hisanori Mishima's web page for some nontrivial solutions. - M. F. Hasler, Feb 02 2025
LINKS
Daniel Mondot, Table of n, a(n) for n = 1..10000 (first 748 terms from Jovan Radenkovicc)
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020
FORMULA
A043537(A053061(a(n))) <= 4.
Trivially, a(n) >> n^1.66... where the exponent is log(10)/log(4) (A154155). - Charles R Greathouse IV, Jan 30 2025
EXAMPLE
816 is in the sequence since 816^2 = 665856 and both together use at most 4 distinct digits.
149 is not in the sequence since 149^2 = 22201 and both together use 5 distinct digits.
MATHEMATICA
Select[Range[0, 110], Length[Union @@ IntegerDigits@ {#, #^2}] < 5 &] (* Amiram Eldar, Nov 15 2024 *)
PROG
(Magma)
[n: n in [0..1000000] | #Set(Intseq(n)) le 4 and #Set(Intseq(n) cat Intseq(n^2)) le 4];
(PARI) isok(k) = #Set(concat(digits(k), digits(k^2))) <= 4; \\ Michel Marcus, Nov 15 2024
(PARI) is(n)=my(s=Set(digits(n))); #s<5 && #setunion(Set(digits(n^2)), s)<5 \\ Charles R Greathouse IV, Jan 30 2025
(PARI) is1(n)=#setunion(Set(digits(n^2)), Set(digits(n)))<5
ok(m)=my(d=concat(apply(k->digits(lift(k)), [m, m^2]))
test(d)=my(v=List(), D=10^d); for(n=0, D-1, if(ok(Mod(n, D)), listput(v, n))); Vec(v)
res=test(8); \\ build a list of residues mod 10^8
D=diff(concat(res, res[1]+10^8)); #D
u=List(); for(n=0, 10^7, if(is1(n) && !setsearch(n, res), listput(u, n))); \\ build exceptions
setminus(select(is1, [0..n]), list(n))
list(lim)=my(v=List(u)); forstep(n=0, lim, D, if(is1(n), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jan 30 2025
(Python)
def ok(n): return len(set(str(n)+str(n**2))) <= 4
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Nov 18 2024
CROSSREFS
Supersequence of A136808, A136809, A136816, A136822.
Sequence in context: A247250 A017909 A345201 * A316530 A296864 A124695
KEYWORD
nonn,base
AUTHOR
Jovan Radenkovicc, Nov 15 2024
STATUS
approved