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

1,3

Comments

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)

Links

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

Formula

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

Example

a(21055) = 9427771^2 = 88882866028441, - Zak Seidov, Apr 18 2016
a(21056) = 9427980^2 = 88886806880400. - Robert Israel, Apr 18 2016

Maple

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

Mathematica

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

Crossrefs

Setwise difference A000290 \ A030474.

Cf. A030477.

Sequence in context: A325149 A326707 A352618 * A077355 A077487 A179334

Adjacent sequences: A030473 A030474 A030475 * A030477 A030478 A030479

Keyword

nonn,base

Author

Patrick De Geest

Extensions

Offset corrected by Andrew Howroyd, Aug 11 2024

Status

approved