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 1-9 of 9 results.

A114258 Numbers k such that k^2 contains exactly 2 copies of each digit of k.

Original entry on oeis.org

72576, 406512, 415278, 494462, 603297, 725760, 3279015, 4065120, 4152780, 4651328, 4915278, 4927203, 4944620, 4972826, 4974032, 4985523, 4989323, 5002245, 5016125, 6032970, 6214358, 6415002, 6524235, 7257600, 9883667
Offset: 1

Views

Author

Giovanni Resta, Nov 18 2005

Keywords

Comments

From Chai Wah Wu, Feb 27 2024: (Start)
If k is a term, then k == 0 (mod 9) or k == 2 (mod 9) (see A370676).
First decimal digit of each term is 3 or larger. (End)

Examples

			72576 is in the sequence since its square 5267275776 contains four 7's, two 2's, two 5's and two 6's.

Links

Crossrefs

Cf. A114259, A114260, A114261, A199630.
Cf. A061656, A061657, A061658, A061659, A061660, A061661, A061662, A061663, A061664.

Programs

  • Python
    from math import isqrt
from itertools import count, islice
def A114258_gen(): # generator of terms
    for l in count(1):
        a = isqrt(10**((l<<1)-1))
        if (a9:=a%9):
            a -= a9
        for b in range(a,10**l,9):
            for c in (0,2):
                k = b+c
                if sorted(str(k)*2)==sorted(str(k**2)):
                    yield k
A114258_list = list(islice(A114258_gen(),20)) # Chai Wah Wu, Feb 27 2024

A061656 Numbers k > 1 such that, in base 2, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

53, 106, 211, 212, 397, 403, 417, 419, 422, 424, 437, 441, 459, 781, 794, 801, 806, 817, 833, 834, 838, 839, 841, 844, 848, 865, 874, 882, 885, 918, 979, 1481, 1549, 1562, 1565, 1571, 1573, 1585, 1588, 1589, 1602, 1612, 1613, 1634, 1637, 1665, 1666, 1667, 1668
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			53 = 110101_2 and 53^2 = 101011111001_2.

Links

Crossrefs

Cf. A061657, A061658, A061659, A061660, A061661, A061662, A061663, A114258, A061664.

Programs

  • Maple
    p:= n-> add(x^i, i=convert(n, base, 2)):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 10 2015
  • Mathematica
    b2pQ[n_]:=Module[{bn=IntegerDigits[n,2],b2n=IntegerDigits[n^2,2], cbn0, cb2n0}, cbn0=Count[bn,0];cb2n0=Count[b2n,0];cbn0>0&&cb2n0>0 && Count[ bn,1]/cbn0==Count[b2n,1]/cb2n0]; Select[Range[1700],b2pQ] (* Harvey P. Dale, Jan 25 2012 *)
  • Python
    from fractions import Fraction
from itertools import count, islice
def f(i, j):
    bi, bj = bin(i)[2:], bin(j)[2:]
    pi = [Fraction(bi.count(d), len(bi)) for d in "01"]
    pj = [Fraction(bj.count(d), len(bj)) for d in "01"]
    return pi == pj
def ok(n): return f(n, n**2)
print([k for k in range(2, 1700) if ok(k)]) # Michael S. Branicky, Feb 27 2023

Extensions

Offset changed to 1 by Alois P. Heinz, May 10 2015

A061657 Numbers k > 1 such that, in base 3, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

184, 196, 464, 500, 544, 550, 552, 584, 588, 622, 626, 670, 706, 1392, 1436, 1472, 1500, 1552, 1632, 1650, 1654, 1656, 1744, 1752, 1764, 1866, 1868, 1878, 1978, 2010, 2030, 2116, 2118, 3922, 4136, 4176, 4308, 4388, 4416, 4500, 4656, 4732, 4756, 4896, 4900
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			184 = 20211_3 and 184^2 = 1201102221_3.

Links

Crossrefs

Cf. A061656, A061658, A061659, A061660, A061661, A061662, A061663, A114258, A061664.

Programs

  • Maple
    p:= n-> add(x^i, i=convert(n, base, 3)):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 10 2015

Extensions

Offset changed to 1 by Alois P. Heinz, May 10 2015

A061658 Numbers k > 1 such that, in base 4, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

45, 165, 180, 657, 660, 720, 882, 2165, 2193, 2331, 2625, 2628, 2640, 2880, 3362, 3470, 3528, 3606, 3683, 3825, 8285, 8294, 8337, 8381, 8477, 8493, 8519, 8525, 8660, 8721, 8772, 8817, 9069, 9282, 9324, 9479, 9507, 9869, 9969, 10185, 10349, 10353, 10500, 10512
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			45 = 231_4 and 45^2 = 133221_4.

Links

Crossrefs

Cf. A061656, A061657, A061659, A061660, A061661, A061662, A061663, A114258, A061664.

Programs

  • Maple
    p:= n-> add(x^i, i=convert(n, base, 4)):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 10 2015

Extensions

Offset changed to 1 by Alois P. Heinz, May 10 2015

A061659 Numbers k > 1 such that, in base 5, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

9726, 10512, 10904, 12702, 12886, 13766, 13898, 14008, 15086, 37644, 41156, 42006, 42106, 44048, 44448, 44578, 44756, 44826, 48630, 48664, 49482, 49626, 49652, 49676, 51704, 52560, 52582, 54262, 54520, 54982, 57922, 59672, 60778, 63510, 64430, 64736, 65432
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			9726 = 302401_5 and 9726^2 = 143204020301_5.

Links

Crossrefs

Cf. A061656, A061657, A061658, A061660, A061661, A061662, A061663, A114258, A061664.

Programs

  • Maple
    p:= n-> add(x^i, i=convert(n, base, 5)):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 10 2015

Extensions

Offset changed to 1 by Alois P. Heinz, May 10 2015

A061660 Numbers k > 1 such that, in base 6, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

3697, 3940, 4802, 5845, 5905, 21127, 21715, 22182, 22867, 22897, 23380, 23640, 24367, 26815, 28812, 28910, 32192, 33705, 33815, 35000, 35065, 35070, 35430, 35977, 37082, 37712, 40277, 44535, 122915, 125947, 126762, 128350, 129670, 130290, 133092, 134397
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			3697 = 25041_6 and 3697^2 = 1204540521_6.

Links

Crossrefs

Cf. A061656, A061657, A061658, A061659, A061661, A061662, A061663, A114258, A061664.

Programs

  • Maple
    p:= n-> add(x^i, i=convert(n, base, 6)):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while p(k)*2<>p(k^2) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 10 2015

Extensions

Offset changed to 1 by Alois P. Heinz, May 10 2015

A061661 Numbers k > 1 such that, in base 7, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

30, 210, 1470, 10290, 10652, 11636, 53544, 53730, 59130, 64896, 72030, 74564, 81452, 93510, 94356, 94400, 97038, 113754, 348540, 364016, 374808, 375204, 376110, 378642, 413910, 420848, 454272, 476352, 479642, 483428, 485460, 488232, 503648, 504210, 516302
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			30 = 42_7 and 30^2 = 2424_7.

Links

Crossrefs

Cf. A061656, A061657, A061658, A061659, A061660, A061662, A061663, A114258, A061664.

Extensions

More terms from Naohiro Nomoto, Oct 04 2001
Offset changed to 1 by Alois P. Heinz, May 10 2015

A061662 Numbers k > 1 such that, in base 8, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

266, 273, 282, 2058, 2065, 2128, 2184, 2256, 13771, 14931, 16394, 16401, 16450, 16464, 16513, 16520, 16634, 16674, 16898, 17024, 17409, 17472, 18048, 24852, 110168, 119448, 122297, 131082, 131089, 131138, 131152, 131201, 131208, 131600
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			266 = 412_8 and 266^2 = 212144_8.

Links

Crossrefs

Cf. A061656, A061657, A061658, A061659, A061660, A061661, A061663, A114258, A061664.

Extensions

Offset changed to 1 by Alois P. Heinz, May 10 2015

A061663 Numbers k > 1 such that, in base 9, k and k^2 contain the same digits in the same proportion.

Original entry on oeis.org

2890, 26010, 188722, 234090, 235954, 260416, 390688, 484094, 511234, 1621474, 1651654, 1698498, 1953842, 2022214, 2106810, 2123586, 2202230, 2283388, 2296028, 2343744, 2485838, 2496508, 2559962, 2595386, 2627210, 2841118, 3091018, 3118760, 3309540, 3334316, 3483208
Offset: 1

Views

Author

Erich Friedman, Jun 16 2001

Keywords

Examples

			2890 = 3861_9 and 2890^2 = 16638831_9.

Links

Crossrefs

Cf. A061656, A061657, A061658, A061659, A061660, A061661, A061662, A114258, A061664.

Extensions

More terms from Naohiro Nomoto, Oct 04 2001
Offset changed to 1 by Alois P. Heinz, May 10 2015
Terms a(13)-a(31) and beyond from Seiichi Manyama, Aug 20 2023
Showing 1-9 of 9 results.

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