A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

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

Offset: 1

Author

Bernard Schott, Oct 27 2019

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A052040, A104315, A134844.

Cf. A323780, A328782, A328783.

Programs

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 *)