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.

Previous Showing 31-39 of 39 results.

A370872 Positive integers m such that c(0) > c(1) >= c(2), where c(k) = number of k's in the ternary representation of m.

Original entry on oeis.org

9, 27, 29, 33, 45, 55, 57, 63, 81, 82, 83, 84, 87, 90, 99, 108, 135, 163, 165, 171, 189, 243, 244, 245, 246, 248, 249, 250, 252, 254, 258, 261, 262, 264, 270, 272, 276, 288, 297, 298, 300, 306, 324, 326, 330, 342, 378, 405, 406, 408, 414, 432, 487, 489, 490
Offset: 1

Views

Author

Clark Kimberling, Mar 13 2024

Keywords

Examples

			The ternary representation of 84 is 10010, for which c(0)=3 > c(1)=2 >= c(2)=0.

Crossrefs

Cf. A007089, A077267, A062756, A081603.
Cf. A370870, A370871, A370873.

Programs

  • Mathematica
    Select[Range[1000], DigitCount[#, 3, 0] > DigitCount[#, 3, 1] >= DigitCount[#, 3, 2] &]

A370873 Positive integers m such that c(0) >= c(1) >= c(2), where c(k) = number of k's in the ternary representation of m.

Original entry on oeis.org

3, 9, 11, 15, 19, 21, 27, 28, 29, 30, 33, 36, 45, 55, 57, 63, 81, 82, 83, 84, 86, 87, 88, 90, 92, 96, 99, 100, 102, 108, 110, 114, 126, 135, 136, 138, 144, 163, 165, 166, 171, 172, 174, 189, 190, 192, 198, 243, 244, 245, 246, 247, 248, 249, 250, 252, 253
Offset: 1

Views

Author

Clark Kimberling, Mar 13 2024

Keywords

Examples

			The ternary representation of 84 is 10010, for which c(0)=3 >= c(1)=2 >= c(2)=0.

Crossrefs

Cf. A007089, A077267, A062756, A081603.
Cf. A370870, A370871, A370872.

Programs

  • Mathematica
    Select[Range[1000], DigitCount[#, 3, 0] >= DigitCount[#, 3, 1] >= DigitCount[#, 3, 2] &]

A206427 Square array 2^(m-1)*(3^n+1), read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 5, 4, 4, 14, 10, 8, 8, 41, 28, 20, 16, 16, 122, 82, 56, 40, 32, 32, 365, 244, 164, 112, 80, 64, 64, 1094, 730, 488, 328, 224, 160, 128, 128, 3281, 2188, 1460, 976, 656, 448, 320, 256, 256, 9842, 6562, 4376, 2920, 1952, 1312, 896, 640, 512, 512
Offset: 0

Views

Author

Marcus Jaiclin, Feb 07 2012

Keywords

Comments

Rectangular array giving the number of 1's in any row of Pascal's Triangle (mod 3) whose row number has exactly m 1's and n 2's in its ternary expansion (listed by antidiagonals).
a(m,n) is independent of the number of zeros in the ternary expansion of the row number.
a(m,n) gives a non-recursive formula for A206424.

Examples

			Initial 5 X 5 block of entries (upper corner is (m,n)=(0,0), m increases down, n increases across):
1    2    5   14   41
2    4   10   28   82
4    8   20   56  164
8   16   40  112  328
16  32   80  224  656
Pascal's Triangle (mod 3), row numbers in ternary:
1     <=  Row 0, m = 0, n = 0, 2^(-1)(3^0 + 1) = #1's = 1
1 1     <=  Row 1, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2
1 2 1     <=  Row 2, m = 0, n = 1, 2^(-1)(3^1 + 1) = #1's = 2
1 0 0 1     <=  Row 10, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2
1 1 0 1 1     <=  Row 11, m = 2, n = 0, 2^1(3^0 + 1) = #1's = 4
1 2 1 1 2 1     <=  Row 12, m = 1, n = 1, 2^0(3^1 + 1) = #1's = 4
1 0 0 2 0 0 1     <=  Row 20, m = 0, n = 1, 2^(-1)(3^1 + 1) = #1's = 2
1 1 0 2 2 0 1 1     <=  Row 21, m = 1, n = 1, 2^0(3^1 + 1) = #1's = 4
1 2 1 2 1 2 1 2 1     <=  Row 22, m = 0, n = 2, 2^(-1)(3^2 + 1) = #1's = 5
1 0 0 0 0 0 0 0 0 1     <=  Row 100, m = 1, n = 0, 2^0(3^0 + 1) = #1's = 2

Links

Crossrefs

Cf. A206428, A206424, A227428, A083093, A077267, A062756, A081603.
Cf. A062296, A006047, A007318.

Formula

a(m, n) = 2^(m - 1)(3^n + 1).

A206428 Rectangular array, a(m,n) = 2^(m-1)*(3^n-1), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 13, 8, 4, 0, 40, 26, 16, 8, 0, 121, 80, 52, 32, 16, 0, 364, 242, 160, 104, 64, 32, 0, 1093, 728, 484, 320, 208, 128, 64, 0, 3280, 2186, 1456, 968, 640, 416, 256, 128, 0, 9841, 6560, 4372, 2912, 1936, 1280, 832, 512, 256, 0
Offset: 0

Views

Author

Marcus Jaiclin, Feb 07 2012

Keywords

Comments

Number of 2's in any row of Pascal's triangle (mod 3) whose row number has exactly m 1's and n 2's in its ternary expansion.
a(m,n) is independent of the number of zeros in the ternary expansion of the row number.
a(m,n) gives a non-recursive formula for A227428.

Examples

			Initial 5 X 5 block of array (upper left corner is (0,0), row index m, column index n):
0    1    4   13   40
0    2    8   26   80
0    4   16   52  160
0    8   32  104  320
0   16   64  208  640
Pascal's Triangle (mod 3), row numbers in ternary:
1     <= Row 0, m=0, n=0, 2^(-1)(3^0-1) = #2's = 0
1 1     <= Row 1, m=1, n=0, 2^0(3^0-1) = #2's = 0
1 2 1     <= Row 2, m=0, n=1, 2^(-1)(3^1-1) = #2's = 1
1 0 0 1     <= Row 10, m=1, n=0, 2^0(3^0-1) = #2's = 0
1 1 0 1 1     <= Row 11, m=2, n=0, 2^1(3^0-1) = #2's = 0
1 2 1 1 2 1     <= Row 12, m=1, n=1, 2^0(3^1-1) = #2's = 2
1 0 0 2 0 0 1     <= Row 20, m=0, n=1, 2^(-1)(3^1-1) = #2's = 1
1 1 0 2 2 0 1 1     <= Row 21, m=1, n=1, 2^0(3^1-1) = #2's = 2
1 2 1 2 1 2 1 2 1     <= Row 22, m=0, n=2, 2^(-1)(3^2-1) = #2's = 4
1 0 0 0 0 0 0 0 0 1     <= Row 100, m=1, n=0, 2^0(3^0-1) = #2's = 0

Links

Crossrefs

Cf. A206427, A206424, A227428, A083093, A077267, A062756, A081603.
Cf. A062296, A006047, A007318.

A291771 Filter based on runlengths of 0-digits in base-3 expansion of n: a(n) = A278222(A291770(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 8, 4, 4, 6, 2, 2, 6, 2, 2, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 8, 4, 4, 6, 2, 2, 6, 2, 2, 4, 2, 2, 2, 1, 1, 2, 1, 1, 4, 2, 2, 2, 1, 1, 2, 1, 1, 16, 8, 8, 12, 4, 4, 12, 4, 4, 12, 6, 6, 6, 2, 2, 6, 2, 2, 12, 6, 6, 6, 2, 2, 6
Offset: 1

Views

Author

Antti Karttunen, Sep 11 2017

Keywords

Links

Crossrefs

Cf. A007089, A077267, A278222, A291770, A290091, A290092, A290093, A290094.

Programs

  • Mathematica
    f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[FromDigits[IntegerDigits[#, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2], 1, 1] &, 96] (* Michael De Vlieger, Sep 11 2017 *)

Formula

a(n) = A278222(A291770(n)).

A365279 Inverse permutation to A365278.

Original entry on oeis.org

0, 1, 3, 2, 7, 15, 6, 31, 63, 4, 5, 11, 14, 127, 255, 30, 511, 1023, 12, 13, 27, 62, 2047, 4095, 126, 8191, 16383, 8, 9, 19, 10, 23, 47, 22, 95, 191, 28, 29, 59, 254, 32767, 65535, 510, 131071, 262143, 60, 61, 123, 1022, 524287, 1048575, 2046, 2097151, 4194303
Offset: 0

Views

Author

Rémy Sigrist, Aug 31 2023

Keywords

Examples

			A365278(42) = 273, hence a(273) = 42.

Links

Crossrefs

Cf. A023416, A032924, A077267, A365278.

Programs

  • PARI
    See Links section.

Formula

a(3*n) = 2*a(n).
a(A032924(k)) = 2^k - 1 for any k > 0.
A023416(a(n)) = A077267(n).

A039000 Numbers whose base-3 representation has the same number of 0's and 1's.

Original entry on oeis.org

2, 3, 8, 11, 15, 19, 21, 26, 28, 30, 35, 36, 47, 51, 59, 61, 65, 69, 73, 75, 80, 86, 88, 92, 96, 100, 102, 107, 110, 114, 126, 136, 138, 143, 144, 155, 159, 166, 172, 174, 179, 185, 187, 190, 192, 197, 198, 209, 213, 221, 223, 227, 231, 235, 237, 242, 247, 253
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A007089, A077267, A062756.

Programs

  • Mathematica
    Select[Range[300],DigitCount[#,3,0]==DigitCount[#,3,1]&] (* Harvey P. Dale, Sep 04 2020 *)

A096405 Numbers having an equal number of zeros in their binary and ternary representations.

Original entry on oeis.org

0, 1, 6, 7, 9, 11, 28, 45, 47, 57, 59, 60, 61, 62, 81, 83, 90, 91, 93, 95, 99, 108, 109, 110, 117, 119, 123, 171, 183, 190, 207, 219, 222, 223, 248, 303, 315, 378, 379, 381, 383, 405, 407, 411, 414, 415, 423, 447, 459, 477, 479, 488, 490, 497, 498, 499, 502, 508
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 07 2004

Keywords

Comments

A023416(a(n)) = A077267(a(n)).

Examples

			n=60: A007088(60)='111100', A007089(60)='2020': both containing two zeros, therefore 60 is a term.

Links

Crossrefs

Cf. A023416, A077267.

Programs

  • Maple
    filter:= n -> numboccur(0,convert(n,base,2))=numboccur(0,convert(n,base,3)):
select(filter, [$0..1000]); # Robert Israel, Jan 16 2018
  • Mathematica
    Select[Range[0,600],DigitCount[#,2,0]==DigitCount[#,3,0]&] (* Harvey P. Dale, Feb 02 2012 *)

A081602 Number of 0's in ternary representation of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1
Offset: 0

Views

Author

Reinhard Zumkeller, Mar 23 2003

Keywords

Comments

Duplicate of A077267.
Previous Showing 31-39 of 39 results.

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