A134845 -id:A134845 - cp's OEIS frontend

A134844 Numbers k such that k contains no zero but k^2 contains at least one zero.

32, 33, 45, 47, 48, 49, 51, 52, 53, 55, 64, 71, 78, 84, 95, 97, 98, 99, 138, 142, 143, 144, 145, 147, 148, 149, 151, 152, 153, 155, 174, 175, 176, 179, 195, 197, 198, 199, 217, 224, 225, 226, 241, 243, 245, 246, 247, 248, 249, 251, 252, 253, 255, 257, 259, 265

Offset: 1

Artur Jasinski, Nov 13 2007

Contains all numbers with no zero that start with 32. - Robert Israel, Jun 25 2019

			a(1)=32 because 1024 = 32^2.

Robert Israel, Table of n, a(n) for n = 1..10000
J. Browkin, Groebner basis.

Cf. A052040, A134843, A134845.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  if member(0,L) then return false fi;
  L:= convert(n^2,base,10);
  member(0,L)
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 25 2019

a = {}; Do[Do[Do[k = 100s + 10n + m; w = IntegerDigits[k^2]; If[MemberQ[w, 0], AppendTo[a, k]], {n, 1, 9}], {m, 1, 9}], {s, 0, 9}]; Union[a]
Select[Range[300],DigitCount[#,10,0]==0&&DigitCount[#^2,10,0]>0&] (* Harvey P. Dale, Mar 20 2012 *)

Definition clarified by Harvey P. Dale, Mar 20 2012

A134843 Numbers n not divisible by 10 such that n^2 contains a 0.

Original entry on oeis.org

32, 33, 45, 47, 48, 49, 51, 52, 53, 55, 64, 71, 78, 84, 95, 97, 98, 99, 101, 102, 103, 104, 105, 138, 142, 143, 144, 145, 147, 148, 149, 151, 152, 153, 155, 174, 175, 176, 179, 195, 197, 198, 199, 201, 202, 203, 205, 217, 224, 225, 226, 241, 243, 245, 246, 247

Offset: 1

Artur Jasinski, Nov 13 2007

			a(1)=32 because 1024=32^2

Harvey P. Dale, Table of n, a(n) for n = 1..1000
J. Browkin, Groebner basis.

Cf. A134844, A134845.

a = {}; Do[Do[Do[k = 100s + 10n + m; w = IntegerDigits[k^2]; If[MemberQ[w, 0], AppendTo[a, k]], {m, 1, 9}], {n, 0, 9}], {s, 0, 9}]; Union[a]
Select[Range[300],Mod[#,10]!=0&&DigitCount[#^2,10,0]>0&] (* Harvey P. Dale, Nov 15 2023 *)

A134846 Smallest number k containing no zero digit such that k^2 contains exactly n zeros.

Original entry on oeis.org

32, 245, 448, 3747, 24495, 62498, 248998, 2449552, 6393747, 6244998, 244949995, 498998998, 2449489753, 24498999998, 28284271249, 248997999998, 498998999999, 4989989999997, 24899979999998

Offset: 1

Artur Jasinski, Nov 13 2007

The corresponding squares are in A134847.
Browkin (see link, p. 29) gives a number without zero digits whose square has 26 zeros: 4472135954999579392819^2 = 20000000000000000000005837591200400708766761. However, he does not claim that it is the smallest such number, so a(26) <= 4472135954999579392819.
Indeed, there are much smaller candidates for a(26), such as 489899998999999999. We also have a(20) <= 49899989999999 and a(21) <= 498998998999998. - Giovanni Resta, Jun 28 2019

			a(1) = 32 because 32 is the smallest number without zero digits whose square has exactly one zero: 1024.

Jerzy Browkin, Groebner basis (in Polish)

Cf. A134843, A134844, A134845, A134847.

Edited and a(11), a(12), a(13) added by Klaus Brockhaus, Nov 20 2007
a(14)-a(15) from Lars Blomberg, Jun 25 2011
a(16)-a(19) from Giovanni Resta, Jun 28 2019

A134847 Smallest square number with exactly n zero digits whose square root does not contain any zero digits.

Original entry on oeis.org

1024, 60025, 200704, 14040009, 600005025, 3906000004, 62000004004, 6000305000704, 40880000700009, 39000000020004, 60000500050500025, 249000000005004004, 6000000050052001009, 600201000902004000004, 800000000087008020001, 62000004003004008000004

Offset: 1

Artur Jasinski, Nov 13 2007

a(26) <= 20000000000000000000005837591200400708766761.
For corresponding square roots and more information see A134846.
a(26) <= 240002009020200000020200002000000001. - Giovanni Resta, Jun 28 2019

			a(1) = 1024 because 1024 is the smallest square with one zero whose square root has no zero: 32.

Giovanni Resta, Table of n, a(n) for n = 1..19

Cf. A134843, A134844, A134845, A134846.

Edited and a(11)-a(13) added by Klaus Brockhaus, Nov 20 2007

a(14)-a(16) from Giovanni Resta, Jun 28 2019

A134848 Number of digits in A134847(n).

Original entry on oeis.org

4, 5, 6, 8, 9, 10, 11, 13, 14, 14, 17, 18, 19, 21, 21, 23, 24, 26, 27

Offset: 1

Artur Jasinski, Nov 13 2007

a(26)=44

J. Browkin, Groebner basis.

Cf. A134843, A134844, A134845, A134846, A134847.

a(11)-a(15) from Lars Blomberg, Jun 25 2011

A134849 Number of nonzero digits in A134847(n).

Original entry on oeis.org

3, 3, 3, 4, 4, 4, 4, 5, 5, 4, 6, 6, 6, 7, 6, 7, 7, 8, 8

Offset: 1

Artur Jasinski, Nov 13 2007

J. Browkin, Groebner basis (in Polish).

Cf. A134843, A134844, A134845, A134846, A134847.

a(11)-a(15) from Lars Blomberg, Jun 25 2011

a(16)-a(19) (using A134847 bfile) from Michel Marcus, Aug 17 2020

A135215 Maximal number of zero digits in square of number with n digits and without zero digits.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 10, 10, 12, 13, 15

Offset: 1

Artur Jasinski, Nov 23 2007

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849.

(*For a(7) *) mx = 0; Do[Do[Do[Do[Do[Do[Do[k = 10^6b + 10^5q + 10^4r + 10^3p + 10^2s + 10n + m; w = IntegerDigits[k^2]; ile = 0; Do[If[w[[t]] == 0, ile = ile + 1; If[ile > mx, mx = ile]], {t, 1, Length[w]}], {m, 1, 9}], {n, 1, 9}], {s, 1, 9}], {p, 1, 9}], {r, 1, 9}], {q, 1, 9}], {b, 1, 9}]; mx

a(8)-a(11) from Lars Blomberg, Jun 26 2011

A135217 a(n) = smallest number with n+1 digits and without zero digits whose squares have the maximal number of zero digits = A135215(n+1).

Original entry on oeis.org

32, 448, 3747, 62498, 248998, 6244998, 31623251, 498998998, 2449489753, 28284271249

Offset: 2

Artur Jasinski, Nov 23 2007

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135216, A135219.

a(8)-a(11) from Lars Blomberg, Jun 26 2011

A135219 a(n) = largest number with n+1 digits and without zero digits whose squares have maximal number of zero digits = A135215(n+1).

Original entry on oeis.org

99, 949, 9798, 62498, 997998, 6244998

Offset: 2

Artur Jasinski, Nov 23 2007

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135216, A135217.

A135216 a(n)= number of numbers with n+1 digits and without zero digits whose squares have maximal number of zero digits = A135215(n+1).

Original entry on oeis.org

18, 3, 13, 1, 7, 1

Offset: 1

Artur Jasinski, Nov 23 2007

			a(1)=18 because we have 18 numbers with 2 digits and without zero digit whose square have maximal possible value 1 zero: 32, 33, 45, 47, 48, 49, 51, 52, 53, 55, 64, 71, 78, 84, 95, 97, 98, 99.
a(2)=3 because we have 3 numbers with 3 digits and without zero digit whose square have maximal possible value 3 zeros: 448, 548, 949.
a(3)=13 because we have 13 numbers with 4 digits and without zero digit whose square have maximal possible value 4 zeros: 3747, 3751, 4899, 6245, 6249, 6253, 7746, 7747, 7749, 7751, 7753, 9747, 9798.
a(4)=1 because we have only one number with 5 digits and without zero digit whose square have maximal possible value 6 zeros: 62498.
a(5)=7 because we have 7 numbers with 6 digits and without zero digit whose square have maximal possible value 7 zeros: 248998, 316245, 489898, 498999, 781249, 948951, 997998.
a(6)=1 because we have only one number with 7 digits and without zero digit whose square have maximal possible value 10 zeros: 6244998.

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135217, A135219.

(*For a(7) *) c = 0; mx = 10; Do[Do[Do[Do[Do[Do[Do[k = 10^6b + 10^5q + 10^4r + 10^3p + 10^2s + 10n + m; w = IntegerDigits[k^2]; ile = 0; Do[If[w[[t]] == 0, ile = ile + 1], {t, 1, Length[w]}]; If[ile == mx, c = c + 1], {m, 1, 9}], {n, 1, 9}], {s, 1, 9}], {p, 1, 9}], {r, 1, 9}], {q, 1, 9}], {b, 1, 9}]; c (*Artur Jasinski*)

cp's OEIS Frontend

A134844 Numbers k such that k contains no zero but k^2 contains at least one zero.

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A134843 Numbers n not divisible by 10 such that n^2 contains a 0.

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A134846 Smallest number k containing no zero digit such that k^2 contains exactly n zeros.

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A134847 Smallest square number with exactly n zero digits whose square root does not contain any zero digits.

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A134848 Number of digits in A134847(n).

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A134849 Number of nonzero digits in A134847(n).

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A135215 Maximal number of zero digits in square of number with n digits and without zero digits.

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A135217 a(n) = smallest number with n+1 digits and without zero digits whose squares have the maximal number of zero digits = A135215(n+1).

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A135219 a(n) = largest number with n+1 digits and without zero digits whose squares have maximal number of zero digits = A135215(n+1).

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A135216 a(n)= number of numbers with n+1 digits and without zero digits whose squares have maximal number of zero digits = A135215(n+1).

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