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.

Showing 1-3 of 3 results.

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

Views

Author

Bernard Schott, Oct 27 2019

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

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

Programs

  • Magma
    nz:=func; [k:k in [0..5000] | nz(k) eq nz(k^2)]; // Marius A. Burtea, Dec 21 2020
  • Maple
    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
  • Mathematica
    Select[Range[0, 5000], Equal @@ Total /@ Sign@ IntegerDigits[{#, #^2}] &] (* Giovanni Resta, Feb 27 2020 *)
  • PARI
    isok(k) = hammingweight(digits(k)) == hammingweight(digits(k^2)); \\ Michel Marcus, Dec 22 2020

Extensions

More terms from Alois P. Heinz, Oct 27 2019

A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

Original entry on oeis.org

0, 104, 105, 203, 205, 302, 303, 305, 402, 403, 405, 504, 505, 506, 507, 508, 509, 601, 602, 603, 605, 609, 701, 702, 703, 705, 708, 709, 801, 802, 803, 805, 901, 902, 903, 905, 906, 1006, 1007, 1008, 1009, 1011, 1012, 1013, 1014, 1016, 1017, 1018, 1019, 1021
Offset: 1

Views

Author

Bernard Schott, Oct 28 2019

Keywords

Examples

			703 and 494209 = 703^2 both have one zero digit in their decimal expansion.

Links

Crossrefs

Equals A328781 \ A052040.
Cf. A104315, A134844, A328780, A328783.

Programs

  • Maple
    f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and x=y)(f(n), f(n^2)):
select(q, [$0..1030])[];  # Alois P. Heinz, Oct 28 2019
  • Mathematica
    Select[Range[0, 1100], DigitCount[#, 10, 0] == DigitCount[#^2, 10, 0] > 0 &] (* Giovanni Resta, Feb 27 2020 *)

Extensions

More terms from Alois P. Heinz, Oct 28 2019

A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 401, 410, 420, 430, 440, 450, 460, 470
Offset: 1

Views

Author

Bernard Schott, Oct 28 2019

Keywords

Comments

This sequence is one of the three sequences whose numbers k and k^2 don't contain the same number of 0, the two others are A104315 and A134844.

Examples

			201 and 40401 = 201^2 have both at least one zero but not the same number of 0 in their decimal expansion, hence, 201 is a term.

Links

Crossrefs

Cf. A052040, A104315, A134844, A328780, A328781, A328782.

Programs

  • Maple
    f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and y>0 and x<>y)(f(n), f(n^2)):
select(q, [$0..500])[];  # Alois P. Heinz, Oct 28 2019
  • Mathematica
    Select[Range[0, 470], (x = DigitCount[#, 10, 0]) > 0 && (y = DigitCount[ #^2, 10, 0]) > 0 && x != y &] (* Giovanni Resta, Feb 27 2020 *)

Extensions

More terms from Alois P. Heinz, Oct 28 2019
Showing 1-3 of 3 results.

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