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 1-5 of 5 results.

A226722 Positions of the numbers 2^n, for n >=0, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 12, 15, 17, 18, 21, 22, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 45, 47, 50, 51, 54, 56, 58, 60, 61, 64, 66, 68, 70, 73, 74, 76, 78, 80, 83, 84, 87, 89, 90, 93, 95, 97, 99, 101, 103, 105, 107, 109, 112, 113, 116, 117, 119, 122, 123, 126
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2013

Keywords

Examples

			The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 2^n for n >= 0 are in positions 1, 2, 4, 6, 8, 11, 12, 15, 17, 18.

Links

Crossrefs

Cf. A123384, A226720, A226723, A226724, A306044.

Programs

  • Mathematica
    z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[n + f[(n-1)*Log[c, b]] + f[(n-1)*Log[d, b]], {n, 1, z}]  (* this sequence *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

Formula

a(n) = n + floor((n-1)*log_3(2)) + floor((n-1)*log_5(2)). [corrected by Jason Yuen, Nov 02 2024]

A226723 Positions of the numbers 3^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

Original entry on oeis.org

3, 7, 10, 13, 16, 20, 23, 26, 30, 32, 36, 40, 42, 46, 49, 52, 55, 59, 62, 65, 69, 72, 75, 79, 82, 85, 88, 92, 94, 98, 102, 104, 108, 111, 114, 118, 121, 124, 127, 131, 133, 137, 141, 144, 147, 150, 154, 157, 160, 164, 166, 170, 174, 176, 180, 183, 186, 189
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2013

Keywords

Examples

			The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 3^n for n >= 1 are in positions 3, 7, 10, 13, 16.

Links

Crossrefs

Cf. A123384, A226720, A226722, A226724, A306044.

Programs

  • Mathematica
    z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

Formula

a(n) = 1 + n + floor(n*log_2(3)) + floor(n*log_5(3)).

A226724 Positions of the numbers 5^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

Original entry on oeis.org

5, 9, 14, 19, 24, 28, 34, 38, 43, 48, 53, 57, 63, 67, 71, 77, 81, 86, 91, 96, 100, 106, 110, 115, 120, 125, 129, 135, 139, 143, 148, 153, 158, 162, 168, 172, 177, 182, 187, 191, 197, 201, 205, 211, 215, 220, 225, 230, 234, 240, 244, 249, 254, 259, 263, 269
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2013

Keywords

Examples

			The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 5^n for n >= 0 are in positions 5, 9, 14.

Links

Crossrefs

Cf. A123384, A226720, A226722, A226723, A306044.

Programs

  • Mathematica
    z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)

Formula

a(n) = 1 + n + floor(n*log_2(5)) + floor(n*log_3(5)).

A286989 Positions of 1 in A286987; complement of A286988.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 89, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105
Offset: 1

Views

Author

Clark Kimberling, May 19 2017

Keywords

Comments

a(n) - a(n-1) is in {1,2} for n>=2, and a(n)/n -> (10 + sqrt(2))/7.
Is this the union of {1} and A226720? - R. J. Mathar, May 21 2017

Examples

			As a word, A286987 = 1101101011010110110101..., in which 1 is in positions 1,2,4,5,7,9,....

Links

Crossrefs

Cf. A080764, A286987, A286988.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> 1, 1 -> {1, 1, 0}}] &, {1}, 11]; (* Sturmian word A080764 *)
w = StringJoin[Map[ToString, s]];
w1 = StringReplace[w, {"111" -> "1"}]
st = ToCharacterCode[w1] - 48 ; (* A286987 *)
Flatten[Position[st, 0]];  (* A286988 *)
Flatten[Position[st, 1]];  (* A286989 *)

A226721 Position of 2^n in the joint ranking of all the numbers 2^j for j>=0 and 5^k for k>=1; complement of A123384.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 53, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95
Offset: 1

Views

Author

Clark Kimberling, Jun 16 2013

Keywords

Examples

			The joint ranking of the powers of 2 and of 5 begins like this: 1, 2, 4, 5, 8, 16, 25, 32, 64, 125, 128, 256, 512.  The numbers 2^n for n >= 1 are in positions 2, 3, 5, 6, 8, 9, 11, 12, 13.

Links

Crossrefs

Cf. A123384, A226720.

Programs

  • Mathematica
    b = 2; c=5; Floor[1 + Range[0, 100]*(1 + Log[b, c])]  (* A123384 *)
Floor[1 + Range[1, 100]*(1 + Log[c, b])]  (* A226721 *)

Formula

a(n) = 1 + A066344(n).
a(n) = 1 + floor(n*(1 + log_5(2))).
Showing 1-5 of 5 results.

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