A030477 -id:A030477 - cp's OEIS frontend

A030476 Squares with property that all even digits occur together and all odd digits occur together.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 144, 196, 225, 289, 324, 400, 441, 484, 576, 625, 784, 841, 900, 1024, 1156, 1444, 1600, 1764, 1936, 2025, 2209, 2401, 2601, 2809, 3136, 3364, 3600, 3844, 4225, 4489, 4624, 5184, 5776, 6084, 6241, 6400

Offset: 1

Patrick De Geest

Among first 22083 terms (up to 10^14), there are 19202 even and 2881 odd terms. Also note that in odd terms, the only odd digit is the last one. - Zak Seidov, Apr 17 2016
From Robert Israel, Apr 18 2016: (Start)
For any k>=1, (10^k-2)^2 is a member of the sequence with the first k-1 digits odd and the last k+1 even, while (2*10^k+2)^2 = 4*10^(2*k)+8*10^k+4 is a member with all digits even, and (2*10^k+1)^2 is an odd member.
If x is an even member, then so is 100*x. (End)

			a(21055) = 9427771^2 = 88882866028441, - _Zak Seidov_, Apr 18 2016
a(21056) = 9427980^2 = 88886806880400. - _Robert Israel_, Apr 18 2016

Zak Seidov, Table of n, a(n) for n = 1..22083

Setwise difference A000290 \ A030474.

Cf. A030477.

filter:= proc(n) local L,evens,odds;
L:= convert(n,base,10) mod 2;
evens:= select(t -> L[t]::even, [$1..nops(L)]);
odds:= select(t -> L[t]::odd, [$1..nops(L)]);
(evens = []) or (odds = []) or (evens[1]>odds[-1]) or (odds[1]>evens[-1])
end proc:
select(filter, [seq(i^2,i=0..100)]); # Robert Israel, Apr 18 2016

Select[Range[0, 80]^2, Function[k, Or[Flatten@ # == k, Flatten@ Reverse@ # == k] &@ GatherBy[k, EvenQ]]@ IntegerDigits@ # &] (* Michael De Vlieger, Apr 17 2016 *)

a(n) = A030477(n)^2. - Andrew Howroyd, Aug 11 2024

Offset corrected by Andrew Howroyd, Aug 11 2024

A030478 Numbers k such that k^3 has property that all even digits occur together and all odd digits occur together.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 10, 11, 12, 13, 15, 17, 19, 20, 22, 24, 26, 31, 33, 34, 35, 39, 40, 42, 46, 52, 54, 58, 59, 71, 72, 80, 82, 91, 92, 93, 95, 100, 104, 110, 120, 124, 131, 135, 137, 150, 152, 169, 172, 173, 174, 175, 179, 200, 202, 211, 220, 240, 258, 259

Offset: 1

Patrick De Geest

From Robert Israel, Oct 27 2024: (Start)
If x is an even member, then so is 10 * x.
2^k + 2 is a member for every k.
(End)

			From _David A. Corneth_, Oct 28 2024: (Start)
12 is in the sequence as 12^3 = 1728 has all even digits occur together and all odd digits occur together. That we can place a bar somewhere between the digits of 1728 so that all odd digits are on one side of the bar and all even digits are on the other side of the bar. In this case that would be 17|28 where all odd digits are on the left of the bar and all even digits are on the other side of the bar.
16 is not in the sequence. To split the even from odd digits in 16^3 = 4096 we would need more than 1 bar, like so: 40|9|6.
321 is not in the sequence as for 321^3 = 33076161 we would need more than two bars to split even from odd. Like so: 33|0|7|6|1|6|1. As the last three digits must be split by more than one bar and 321 has at least three digits no term can end in the last three digits of 321 which means no term can end in 321. Using this idea we can ease the search significantly.  (End)

David A. Corneth, Table of n, a(n) for n = 1..4505 (first 2500 terms from Robert Israel, terms <= 10^10)
David A. Corneth, PARI program

Cf. A030477, A030479.

filter:= proc(n) local L,x,m;
  L:= convert(n^3,base,10) mod 2;
  if n::odd then
    if not member(0,L,m) then return true fi;
    convert(L[m..-1],set) = {0}
  else
    if not member(1,L,m) then return true fi;
    convert(L[m..-1],set) = {1}
  fi
end proc:
select(filter, [$0..1000]);
# Robert Israel, Oct 27 2024

Select[Range[0,300],Length[Split[If[OddQ[#],1,0]&/@IntegerDigits[ #^3]]]<3&] (* Harvey P. Dale, Mar 27 2016 *)

PARI
```
\\ See Corneth link
```

def ok(n):
    s = "".join("0" if d in "02468" else "1" for d in str(n**3))
    return len(set(s.rstrip(s[-1]))) < 2
print([k for k in range(260) if ok(k)]) # Michael S. Branicky, Oct 28 2024

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A030476 Squares with property that all even digits occur together and all odd digits occur together.

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