A134844 -id:A134844 - cp's OEIS frontend

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

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 *)

A134845 Smallest number k not divisible by 10 such that k^2 contains exactly n zeros.

Original entry on oeis.org

32, 101, 448, 1001, 10004, 10001, 100004, 100001, 1000004, 1000001, 10000004, 10000001, 100000004, 100000001, 1000000004, 1000000001

Offset: 1

Artur Jasinski, Nov 13 2007, Nov 16 2007

			a(1)=32 because 32 is smallest number whose square contains exactly one zero.

J. Browkin, Groebner basis.

Cf. A134843, A134844, A134846.

a(9)-a(16) from Arkadiusz Wesolowski, Jan 19 2013

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.

A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

Original entry on oeis.org

0, 1, 2, 3, 10, 20, 30, 100, 200, 245, 247, 249, 251, 253, 283, 300, 448, 548, 949, 1000, 1249, 1253, 1416, 1747, 1749, 1751, 1753, 1755, 2000, 2245, 2247, 2249, 2251, 2253, 2429, 2450, 2451, 2470, 2490, 2498, 2510, 2530, 2647, 2830, 3000, 3747, 3751, 4480, 4899

Offset: 1

Bernard Schott, Oct 27 2019

The idea of this sequence comes from the 1st problem of the 28th British Mathematical Olympiad in 1992 (see the link).
This sequence is infinite because the family of integers {10^k, k >= 0} (A011557) belongs to this sequence.
The numbers m, m + 1, m + 2 where m = 49*10^k - 3, or m = 99*10^k - 3, k >= 3 are terms with all nonzero digits. - Marius A. Burtea, Dec 21 2020

			247^2 = 61009, hence 247 and 61009 both have 3 nonzero digits, 247 is a term.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 57 and 109 (1992)

Giovanni Resta, Table of n, a(n) for n = 1..10000.
British Mathematical Olympiad, 1992 - Problem 1.
Index to sequences related to Olympiads.

Subsequences: A011557, A093136, A093138.
Cf. A052040, A104315, A134844.
Cf. A328781, A328782, A328783.

nz:=func; [k:k in [0..5000] | nz(k) eq nz(k^2)]; // Marius A. Burtea, Dec 21 2020

q:= n->(f->f(n)=f(n^2))(t->nops(subs(0=[][], convert(t, base, 10)))):
select(q, [$0..5000])[];  # Alois P. Heinz, Oct 27 2019

Select[Range[0, 5000], Equal @@ Total /@ Sign@ IntegerDigits[{#, #^2}] &] (* Giovanni Resta, Feb 27 2020 *)

isok(k) = hammingweight(digits(k)) == hammingweight(digits(k^2)); \\ Michel Marcus, Dec 22 2020

More terms from Alois P. Heinz, Oct 27 2019

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

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A134845 Smallest number k not divisible by 10 such that k^2 contains exactly n zeros.

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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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A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

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