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 11-18 of 18 results.

A358793 Lexicographically earliest sequence of positive and unique integers such that 2*Sum_{k = 1..n} a(k) = Sum_{k = 1..n} a(a(k)) for n > 1 and a(1) = 1.

Original entry on oeis.org

1, 3, 7, 5, 10, 8, 14, 16, 11, 20, 22, 13, 26, 28, 17, 32, 34, 19, 38, 40, 23, 44, 46, 25, 50, 52, 29, 56, 58, 31, 62, 64, 35, 68, 70, 37, 74, 76, 41, 80, 82, 43, 86, 88, 47, 92, 94, 49, 98, 100, 53, 104, 106, 55, 110, 112, 59, 116, 118, 61, 122, 124, 65, 128
Offset: 1

Views

Author

Thomas Scheuerle, Dec 01 2022

Keywords

Comments

There is a second version of this sequence possible if we change the definition to a(1) = 2 and a(n) > 1, then the sequence will start 2, 4, 5, 8, 10, 7, 14, ... . It will after this continue in the same way as our actual sequence does (and would also extend the valid range of the recurrence formulas).
Start a(1) = 2 and value 1 allowed is A257794.

Links

Crossrefs

Cf. A002516, A105753, A257794.

Programs

  • Mathematica
    LinearRecurrence[{0, 0, 1, 0, 0, 1, 0, 0, -1}, {1, 3, 7, 5, 10, 8, 14, 16, 11, 20, 22, 13, 26, 28, 17}, 100] (* Paolo Xausa, Jan 23 2025 *)
  • PARI
    a(n) = {my(v = [1, 3, 7, 5, 10, 8]);if(n < 7, v[n], n*(1+min(1, n%3))+(n%3 == 0)+(n%6 == 3))}

Formula

G.f.: x*(1 + 3*x + 7*x^2 + 4*x^3 + 7*x^4 + x^5 + 8*x^6 + 3*x^7 - 4*x^8 + 2*x^9 - x^10 + x^11 - 3*x^12 + x^14)/(1 - x^3 - x^6 + x^9).
a(n) = a(n-3) + a(n-6) - a(n-9) for n >= 16.
a((3*(2*n-1) - (-1)^n)/4) = (3*(2*n-1) - (-1)^n)/2, for n > 3.
a(6*n) = 6*n+1, for n > 1.
a(6*n+3) = 6*n+5, for n > 0.
a(n) = 30*n - 2*a(n-1) - 3*a(n-2) - 3*a(n-3) - 3*a(n-4) - 3*a(n-5) - 2*a(n-6) - a(n-7) - 96, for n > 13.

A007479 Earliest sequence with a(a(a(n))) = 2n.

Original entry on oeis.org

0, 3, 6, 5, 12, 2, 10, 9, 24, 11, 4, 14, 20, 15, 18, 17, 48, 26, 22, 21, 8, 23, 28, 38, 40, 27, 30, 29, 36, 50, 34, 33, 96, 35, 52, 62, 44, 39, 42, 41, 16, 74, 46, 45, 56, 47, 76, 86, 80, 51, 54, 53, 60, 98, 58, 57, 72, 59, 100, 110, 68, 63, 66, 65
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002516.

Formula

a(0) = 0, a(2*n) = 2*a(n), a(6*n+1) = 6*n+3, a(6*n+3) = 6*n+5, a(6*n+5) = 12*n+2. - Sean A. Irvine, Jan 05 2018

Extensions

Missing a(48) = 80 inserted by Sean A. Irvine, Jan 05 2018

A054786 Earliest sequence with a(a(n)) = 6n.

Original entry on oeis.org

0, 2, 6, 4, 18, 7, 12, 30, 9, 48, 11, 60, 36, 14, 78, 16, 90, 19, 24, 102, 21, 120, 23, 132, 108, 26, 150, 28, 162, 31, 42, 174, 33, 192, 35, 204, 72, 38, 222, 40, 234, 43, 180, 246, 45, 264, 47, 276, 54, 50, 294, 52, 306, 55, 288, 318, 57, 336, 59, 348, 66, 62, 366, 64
Offset: 0

Views

Author

Henry Bottomley, Apr 27 2000

Keywords

Links

Crossrefs

Cf. A002516, A002517, A002518, A007379.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = Switch[ Mod[n, 12], 0 | 6, 6*a[n/6], 1 | 3 | 8 | 10, n+1, 2 | 4 | 9 | 11, 6*n-6, 5, n+2, 7, 6*n-12]; Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Dec 20 2011, after formula *)

Formula

a(12n)=6*a(2n), a(12n+1)=12n+2, a(12n+2)=72n+6, a(12n+3)=12n+4, a(12n+4)=72n+18, a(12n+5)=12n+7, a(12n+6)=6*a(2n+1), a(12n+7)=72n+30, a(12n+8)=12n+9, a(12n+9)=72n+48, a(12n+10)=12n+11, a(12n+11)=72n+60.

Extensions

Typo in formula corrected by Reinhard Zumkeller, Jul 23 2010

A054787 Earliest sequence with a(a(n))=7n.

Original entry on oeis.org

0, 2, 7, 4, 21, 6, 35, 14, 9, 56, 11, 70, 13, 84, 49, 16, 105, 18, 119, 20, 133, 28, 23, 154, 25, 168, 27, 182, 147, 30, 203, 32, 217, 34, 231, 42, 37, 252, 39, 266, 41, 280, 245, 44, 301, 46, 315, 48, 329, 98, 51, 350, 53, 364, 55, 378, 63, 58, 399, 60, 413, 62, 427, 392
Offset: 0

Views

Author

Henry Bottomley, Apr 27 2000

Keywords

Links

Crossrefs

Cf. A002516, A002517, A002518, A007379.

Programs

  • Mathematica
    a[n_] := a[n] = Which[ Mod[n, 7] == 0, 7*a[n/7], Mod[n, 7] == 1, n+1, Mod[n, 7] == 2, 7*(n-2)+7, Mod[n, 7] == 3, n+1, Mod[n, 7] == 4, 7*(n-4)+21, Mod[n, 7] == 5, n+1, Mod[n, 7] == 6, 7*(n-6)+35]; a[0] = 0; Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Sep 24 2012 *)

Formula

a(7n)=7*a(n), a(7n+1)=7n+2, a(7n+2)=49n+7, a(7n+3)=7n+4, a(7n+4)=49n+21, a(7n+5)=7n+6, a(7n+6)=49n+35

A054790 Earliest sequence with a(a(n))=10n.

Original entry on oeis.org

0, 2, 10, 4, 30, 6, 50, 8, 70, 11, 20, 90, 13, 120, 15, 140, 17, 160, 19, 180, 100, 22, 210, 24, 230, 26, 250, 28, 270, 31, 40, 290, 33, 320, 35, 340, 37, 360, 39, 380, 300, 42, 410, 44, 430, 46, 450, 48, 470, 51, 60, 490, 53, 520, 55, 540, 57, 560, 59, 580, 500, 62, 610
Offset: 0

Views

Author

Henry Bottomley, Apr 27 2000

Keywords

Links

Crossrefs

Cf. A002516, A002517, A002518, A007379.

Programs

  • Mathematica
    a[0] = 0; a[n_] := Which[m = Mod[n, 20]; m == 0, 10*n-100, m == 9, n+2, m == 10, n+10, m == 11, 10*n-20, MemberQ[ {2, 4, 6, 8, 13, 15, 17, 19}, m], 10*n-10, True, n+1]; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

A054788 Earliest sequence with a(a(n))=8n.

Original entry on oeis.org

0, 2, 8, 4, 24, 6, 40, 9, 16, 56, 11, 80, 13, 96, 15, 112, 64, 18, 136, 20, 152, 22, 168, 25, 32, 184, 27, 208, 29, 224, 31, 240, 192, 34, 264, 36, 280, 38, 296, 41, 48, 312, 43, 336, 45, 352, 47, 368, 320, 50, 392, 52, 408, 54, 424, 57, 72, 440, 59, 464, 61, 480, 63, 496
Offset: 0

Views

Author

Henry Bottomley, Apr 27 2000

Keywords

Links

Crossrefs

Cf. A002516, A002517, A002518, A007379.

Programs

  • Mathematica
    nmax = 63; amax = 8*nmax; t = {{0, a[0] = 0}, {1, a[1] = 2}, {2, a[2]}}; While[ !FreeQ[t, a], t = Table[{n, a[n]}, {n, 0, nmax}]; n = Select[t, !IntegerQ[ #[[2]] ] &, 1][[1, 1]]; t2 = Union[ Flatten[ Append[ Select[ t, IntegerQ[ #[[2]] ] &], n]]]; an = If[n == 2, 8, Select[ Complement[ Range[ Max[t2] ], t2], Mod[#, 8] != 0 &, 1][[1]] ]; a[n] = an; While[ an < amax, an = a[n = an] = 8 n]]; Table[ a[n], {n, 0, nmax}] (* Jean-François Alcover, Jan 11 2012 *)

A054789 Earliest sequence with a(a(n)) = 9n.

Original entry on oeis.org

0, 2, 9, 4, 27, 6, 45, 8, 63, 18, 11, 90, 13, 108, 15, 126, 17, 144, 81, 20, 171, 22, 189, 24, 207, 26, 225, 36, 29, 252, 31, 270, 33, 288, 35, 306, 243, 38, 333, 40, 351, 42, 369, 44, 387, 54, 47, 414, 49, 432, 51, 450, 53, 468, 405, 56, 495, 58, 513, 60, 531, 62, 549
Offset: 0

Views

Author

Henry Bottomley, Apr 27 2000

Keywords

Links

Crossrefs

Cf. A002516, A002517, A002518, A007379, A008585.

Programs

  • Mathematica
    a[0] = 0; a[n_] := Which[m = Mod[n, 18]; m == 0, 9*n-81, m == 9, n+9, MemberQ[ {1, 3, 5, 7, 10, 12, 14, 16}, m], n+1, True, 9*n-9]; Table[ a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 24 2012 *)

A065804 Earliest sequence with a(a(n))=n!.

Original entry on oeis.org

1, 1, 2, 4, 6, 7, 24, 120, 9, 40320, 11, 3628800, 13, 479001600, 15, 87178291200, 17, 20922789888000, 19, 6402373705728000, 21, 2432902008176640000, 23, 1124000727777607680000, 720, 26, 15511210043330985984000000, 28
Offset: 0

Views

Author

Henry Bottomley, Dec 06 2001

Keywords

Examples

			a(3)=4 since any lower value would suggest a(a(3)) was not 3!=6. So a(4)=a(a(3))=3!=6. a(5)=7 since any lower value would prevent a(a(5))=5!. a(6)=a(3!)=a(a(a(3)))=a(a(4))=4!=24. a(7)=a(a(5))=5!=120. a(8)=9 since any lower value would prevent a(a(8))=8!. etc.

Links

Crossrefs

Cf. A002516, A038752, A054048, A054049, A054790, A054791.
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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