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%I #26 Feb 27 2020 07:51:48
%S 0,1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,
%T 28,29,31,34,35,36,37,38,39,41,42,43,44,46,54,56,57,58,59,61,62,63,65,
%U 66,67,68,69,72,73,74,75,76,77,79,81,82,83,85,86,87,88,89,91,92,93,94,96,104,105
%N Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.
%C Inspired by A328780.
%C This sequence is not a duplicate of A052040. The first 72 terms until 96 are exactly the same but a(73) = 104 belongs to this sequence because 104^2 = 10816, but 104 doesn't belong to A052040 because there is one zero digit in the decimal expansion of 104^2.
%C The nonnegative integers that do not belong to this sequence are divided into three sequences:
%C 1) A104315 = A052040 \ {this sequence}: Numbers k such that k contains at least one zero, but k^2 contains no zero (e.g., 106 with 106^2 = 11236).
%C 2) A134844 = Numbers k such that k contains no zero but k^2 contains at least one zero (e.g., 32 with 32^2 = 1024).
%C 3) A328783 = Numbers k such that k and k^2 contain at least one zero but not the same number of zeros (e.g., 101 with 101^2 = 10201).
%C Another sequence is A328782 = {this sequence} \ A052040 which lists the positive integers that have the same positive number of zeros in their decimal expansions as in their squares. The first two examples > 0 are 104 with 104^2 = 10816 and 105 with 105^2 = 11025.
%H Giovanni Resta, <a href="/A328781/b328781.txt">Table of n, a(n) for n = 1..10000</a>
%e 12 and 144 = 12^2 have no digit zero in their decimal representation, so 12 is a term.
%e 203 and 41209 = 203^2 both have one digit zero in their decimal representation, so 203 is also a term.
%p select(t -> numboccur(0, convert(t^2,base,10))=numboccur(0, convert(t,base,10)), [$0..200]); # _Robert Israel_, Oct 27 2019
%t Select[Range[0, 105], Equal @@ Total /@ (1 - Sign@ IntegerDigits[{#, #^2}]) &] (* _Giovanni Resta_, Feb 27 2020 *)
%Y Cf. A052040, A104315, A134844.
%Y Cf. A323780, A328782, A328783.
%K nonn,base
%O 1,3
%A _Bernard Schott_, Oct 27 2019
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