cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A242214 Numbers n not divisible by 10 such that n^3 contains at least one 0.

Original entry on oeis.org

16, 22, 34, 37, 42, 43, 47, 48, 52, 59, 63, 67, 69, 73, 74, 79, 84, 86, 87, 89, 93, 94, 99, 101, 102, 103, 106, 107, 109, 112, 115, 116, 117, 118, 123, 124, 126, 127, 128, 131, 134, 135, 138, 141, 143, 145, 149, 152, 159, 163, 164, 169, 171, 172, 174, 182, 184
Offset: 1

Views

Author

Derek Orr, May 07 2014

Keywords

Comments

n^3 may contain one or more zeros. - Harvey P. Dale, Oct 03 2016

Examples

			16 is not divisible by 10 and 16^3 = 4096, which contains a 0. Thus, 16 is a member of this sequence.

Links

Crossrefs

Cf. A134843.

Programs

  • Mathematica
    Select[Range[200],Mod[#,10]!=0&&DigitCount[#^3,10,0]>0&] (* Harvey P. Dale, Oct 03 2016 *)
  • PARI
    isok(n) = (n % 10) && (vecmin(digits(n^3)) == 0); \\ Michel Marcus, May 10 2014
  • Python
    {print(n) for n in range(10**3) if n%10 != 0 and str(n**3).find("0") > 0}

Extensions

Definition clarified by Harvey P. Dale, Oct 03 2016

A135251 Maximal number of zero digits in square of number with n digits not divisible by 10.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124
Offset: 1

Views

Author

Artur Jasinski, Nov 24 2007

Keywords

Crossrefs

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135217, A135219, A135252, A135253.

Programs

  • Mathematica
    (*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, 0, 9}], {s, 0, 9}], {p, 0, 9}], {r, 0, 9}], {q, 0, 9}], {b, 1, 9}]; mx

Formula

2*n-4 <= a(n) <= 2*n-2 since, if k is an n-digit number not divisible by 10, then k^2 has at most 2*n digits of which the first and last are nonzero; and for n >= 2, the square of the n-digit number 10^(n-1)+1 contains 2*n-4 zeros. It seems likely that a(n) = 2*n-4 for all n >= 4. - Pontus von Brömssen, Jun 09 2025

Extensions

a(8)-a(64) from Pontus von Brömssen, Jun 09 2025

A135252 a(n) = number of numbers with n+1 digits and not divisible by 10 whose squares have maximal number of zero digits = A135251(n+1).

Original entry on oeis.org

18, 3, 24, 11, 10, 11
Offset: 1

Views

Author

Artur Jasinski, Nov 24 2007

Keywords

Examples

			a(1)=18 because we have 18 numbers with 2 digits not divisible by 10 whose squares have maximal possible number of zero digits, namely 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 not divisible by 10 whose square have maximal possible number of zero digits, namely 3 zeros: 448, 548, 949
a(3)=24 because we have 24 numbers with 4 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 4 zeros: 1001, 1002, 1003, 2001, 2002, 3001, 3747, 3751, 4001, 4899, 5001, 5002, 5003, 6245, 6249, 6253, 7746, 7747, 7749, 7751, 7753, 9503, 9747, 9798
a(4)=11 because we have 11 numbers with 5 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 6 zeros: 10001, 10002, 10003, 20001, 20002, 30001, 40001, 50001, 50002, 50003, 62498
a(5)=10 because we have 10 numbers with 6 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 8 zeros: 100001, 100002, 100003, 200001, 200002, 300001, 400001, 500001, 500002, 500003
a(6)=11 because we have 11 numbers with 7 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 10 zeros: 1000001, 1000002, 1000003, 2000001, 2000002, 3000001, 4000001, 5000001, 5000002, 5000003, 6244998

Crossrefs

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135217, A135219, A135251, A135253.

Programs

  • Mathematica
    (* For a(6) *) a = {}; 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; AppendTo[a, k]], {m, 1, 9}], {n, 0, 9}], {s, 0, 9}], {p, 0, 9}], {r, 0, 9}], {q, 0, 9}], {b, 1, 9}]; c

A242559 Zeroless numbers n such that n^3 contains a 0.

Original entry on oeis.org

16, 22, 34, 37, 42, 43, 47, 48, 52, 59, 63, 67, 69, 73, 74, 79, 84, 86, 87, 89, 93, 94, 99, 112, 115, 116, 117, 118, 123, 124, 126, 127, 128, 131, 134, 135, 138, 141, 143, 145, 149, 152, 159, 163, 164, 169, 171, 172, 174, 182, 184, 187, 192, 193, 194, 199, 214, 216
Offset: 1

Views

Author

Derek Orr, May 17 2014

Keywords

Examples

			16 does not contain a 0 and 16^3 = 4096 contains a 0. Thus, 16 is a member of this sequence.

Crossrefs

Subsequence of A242214.
Cf. A052382 (zeroless numbers), A134843.

Programs

  • PARI
    is(n)=Set(digits(n))[1] && Set(digits(n^3))[1]==0 \\ Charles R Greathouse IV, Jul 10 2015
  • Python
    {print(n) for n in range(10**3) if str(n).count("0")==0 and str(n**3).count("0")>0}
Previous Showing 11-14 of 14 results.

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