A137079 -id:A137079 - cp's OEIS frontend

A249915 Numbers k such that the decimal expansions of both k and k^2 have 2 as smallest digit and 6 as largest digit.

255465, 652244, 665256, 2534665, 2536656, 2554262, 6523462, 6524235, 6652242, 23352656, 23354365, 23523462, 23546665, 23565325, 25346665, 25425256, 25624665, 25625465, 65226242, 65234535, 235442656, 254234662, 255465525, 255645525, 256246665, 256254665

Offset: 1

Felix Fröhlich, Apr 21 2015

Chai Wah Wu, Table of n, a(n) for n = 1..2000 (n = 1..42 from Robert Israel).

Cf. A137067, A137079, A137093, A256630, A256631, A256633, A256634, A256708, A256709, A256889, A257197, A257210, A257211, A256601, A257310.

M:= 10:
B:= [[2],[3],[4],[5],[6]]:
A:= NULL:
for d from 2 to M do
  B:= map(b -> seq([op(b), i],i=2..6), B);
  C:= select(b -> max(b)=6 and min(b) = 2, B);
  X:= map(b -> add(b[i]*10^(d-i),i=1..d),C);
  X:= select(proc(x) local L; L:= convert(x^2,base,10); max(L) = 6 and min(L) = 2 end proc, X);
  A:= A, op(X);
od:
A; # Robert Israel, Apr 27 2015

fQ[n_] := Block[{d = DigitCount@ n}, Plus @@ Prepend[Take[d, -4], First@ d] == 0 && d[[2]] > 0 && d[[6]] > 0]; Select[Range@ 2600000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 27 2015 *)

is(n) = vecmin(digits(n))==2 && vecmin(digits(n^2))==2 && vecmax(digits(n))==6 && vecmax(digits(n^2))==6

from itertools import product
A249915_list = []
for l in range(10):
    for a in product('23456', repeat = l):
        for b in ('2', '4', '5', '6'):
            s = ''.join(a)+b
            if '2' in s and '6' in s:
                n = int(s)
                if {'2', '6'} <= set(str(n**2)) <= {'2', '3', '4', '5', '6'}:
                    A249915_list.append(n) # Chai Wah Wu, Apr 29 2015

A378048 Numbers k such that k and k^2 together use at most 4 distinct decimal digits.

Original entry on oeis.org

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

Jovan Radenkovicc, Nov 15 2024

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

			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.

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
Hisanori Mishima, Sporadic tridigital solutions
OEIS index for squares having only given digits

Supersequence of A136808, A136809, A136816, A136822.

Cf. A043537, A053061, A055436, A154155.

[n: n in [0..1000000] | #Set(Intseq(n)) le 4 and #Set(Intseq(n) cat Intseq(n^2)) le 4];

Select[Range[0, 110], Length[Union @@ IntegerDigits@ {#, #^2}] < 5 &] (* Amiram Eldar, Nov 15 2024 *)

isok(k) = #Set(concat(digits(k), digits(k^2))) <= 4; \\ Michel Marcus, Nov 15 2024

is(n)=my(s=Set(digits(n))); #s<5 && #setunion(Set(digits(n^2)),s)<5 \\ Charles R Greathouse IV, Jan 30 2025

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

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

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

cp's OEIS Frontend

A249915 Numbers k such that the decimal expansions of both k and k^2 have 2 as smallest digit and 6 as largest digit.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A378048 Numbers k such that k and k^2 together use at most 4 distinct decimal digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Python

Formula