

A328781


Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.


4



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, 28, 29, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 54, 56, 57, 58, 59, 61, 62, 63, 65, 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
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OFFSET

1,3


COMMENTS

Inspired by A328780.
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.
The nonnegative integers that do not belong to this sequence are divided into three sequences:
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).
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).
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).
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.


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000


EXAMPLE

12 and 144 = 12^2 have no digit zero in their decimal representation, so 12 is a term.
203 and 41209 = 203^2 both have one digit zero in their decimal representation, so 203 is also a term.


MAPLE

select(t > numboccur(0, convert(t^2, base, 10))=numboccur(0, convert(t, base, 10)), [$0..200]); # Robert Israel, Oct 27 2019


MATHEMATICA

Select[Range[0, 105], Equal @@ Total /@ (1  Sign@ IntegerDigits[{#, #^2}]) &] (* Giovanni Resta, Feb 27 2020 *)


CROSSREFS

Cf. A052040, A104315, A134844.
Cf. A323780, A328782, A328783.
Sequence in context: A052382 A055572 A052040 * A069570 A279367 A020731
Adjacent sequences: A328778 A328779 A328780 * A328782 A328783 A328784


KEYWORD

nonn,base


AUTHOR

Bernard Schott, Oct 27 2019


STATUS

approved



