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.

A061987 Number of times n-th distinct value is repeated in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984; also number of times n-th distinct row is repeated in square array T(n,k) = T(n-1,k) + T(n-1,floor(k/2)) + T(n-1,floor(k/3)) with T(0,0) = 1, i.e., in A061980.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 6, 3, 5, 4, 12, 6, 10, 8, 9, 15, 12, 20, 16, 18, 30, 24, 27, 13, 32, 36, 60, 48, 54, 26, 64, 72, 81, 39, 96, 108, 52, 128, 144, 162, 78, 192, 216, 104, 139, 117, 288, 324, 156, 384, 432, 208, 278, 234, 576, 648, 312, 417, 351, 864, 416, 556
Offset: 0

Views

Author

Henry Bottomley, May 24 2001

Keywords

Comments

For n > 0: a(n) = A003586(n+1) - A003586(n) and a(A084791(n)) = A084788(n).
Also number of times A160519(n+1) is repeated in A088468. - Reinhard Zumkeller, May 16 2009
In the 14th century Levi Ben Gerson proved that a(n) > 1 for all n > 6; see A003586, A235365, A235366, A236210. - Jonathan Sondow, Jan 20 2014

Links

Programs

  • Haskell
    import Data.List (group)
a061987 n = a061987_list !! n
a061987_list = map length $ group a061984_list
-- Reinhard Zumkeller, Jan 11 2014

Formula

a(n) = A061986(A061985(n)).

Extensions

More terms from Reinhard Zumkeller, Jun 03 2003

A084789 Increasing gaps between 3-smooth numbers (lower end).

Original entry on oeis.org

1, 4, 9, 12, 18, 36, 81, 108, 162, 256, 288, 324, 512, 576, 648, 768, 864, 1024, 1152, 1296, 1536, 1728, 2304, 2592, 3072, 3456, 4608, 5184, 6912, 9216, 10368, 13824, 20736, 27648, 41472, 82944, 165888, 196608, 221184, 262144, 294912, 331776
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 03 2003

Keywords

Links

Crossrefs

Cf. A003586, A061987, A084788, A084790, A084791.

Programs

  • Mathematica
    s = {}; m = 13; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; s = Union[s]; d = Differences @ s; v = DeleteDuplicates @ FoldList[Max, d]; Map[s[[First@ Position[d, #]]] &, v] //Flatten (* Amiram Eldar, Jan 30 2020 *)

Formula

a(n) = A084790(n) - A084788(n).
a(n) = A003586(A084791(n)).

A084790 Increasing gaps between 3-smooth numbers (upper end).

Original entry on oeis.org

2, 6, 12, 16, 24, 48, 96, 128, 192, 288, 324, 384, 576, 648, 729, 864, 972, 1152, 1296, 1458, 1728, 1944, 2592, 2916, 3456, 3888, 5184, 5832, 7776, 10368, 11664, 15552, 23328, 31104, 46656, 93312, 177147, 209952, 236196, 279936, 314928
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 03 2003

Keywords

Links

Crossrefs

Cf. A003586, A061987, A084788, A084789, A084791.

Programs

  • Mathematica
    s = {}; m = 13; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; s = Union[s]; d = Differences @ s; v = DeleteDuplicates @ FoldList[Max, d]; Map[s[[1 + First@ Position[d, #]]] &, v] //Flatten (* Amiram Eldar, Jan 30 2020 *)

Formula

a(n) = A084788(n) + A084789(n).
a(n) = A003586(A084791(n) + 1).

A084791 Where record gaps between 3-smooth numbers occur.

Original entry on oeis.org

1, 4, 7, 8, 10, 14, 19, 21, 24, 28, 29, 30, 34, 35, 36, 38, 39, 41, 42, 43, 45, 46, 50, 51, 53, 54, 58, 59, 63, 67, 68, 72, 78, 82, 88, 99, 110, 113, 115, 118, 120, 122, 125, 127, 133, 135, 138, 140, 146, 148, 154, 160, 162, 168, 175, 176, 177, 183, 190, 191, 192
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 03 2003

Keywords

Links

Crossrefs

Cf. A003586, A084788, A084789, A084790.

Programs

  • Mathematica
    s = {}; m = 13; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; s = Union[s]; d = Differences@s; v = DeleteDuplicates @ FoldList[Max, d]; Map[First@ Position[d, #] &, v] //Flatten (* Amiram Eldar, Jan 30 2020 *)
Module[{nn=10^7,ts},ts=Select[Range[nn],Max[FactorInteger[#][[;;,1]]]<5&];DeleteDuplicates[Thread[{Range[Length[ts]-1],Differences[ts]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* The program generates the first 58 terms of the sequence. *) (* Harvey P. Dale, Aug 11 2025 *)

Formula

A084788(n) = A061987(a(n)).
A084789(n) = A003586(a(n)).
A084790(n) = A003586(a(n) + 1).

A134361 a(n) = smallest integer >= n which has only prime factors 2 and 3.

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 16, 16, 16, 16, 18, 18, 24, 24, 24, 24, 24, 24, 27, 27, 27, 32, 32, 32, 32, 32, 36, 36, 36, 36, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 54, 54, 54, 54, 54, 54, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 72, 72, 72, 72, 72, 72, 72, 72
Offset: 1

Views

Author

N. J. A. Sloane, Aug 23 2009; corrected Mar 07 2012

Keywords

Comments

Can be used, for example, to derive A084788 and A084790 from A084789. - Charles R Greathouse IV, Jul 09 2021

Crossrefs

Cf. A130916, A134361, A151969.

Programs

  • Mathematica
    With[{pf23=Union[Flatten[Table[Times@@@Tuples[{2,3},n],{n,0,6}]]]}, Flatten[Table[Select[pf23,#>=n&,1],{n,80}]]] (* Harvey P. Dale, Mar 07 2012 *)
  • PARI
    a(n)=my(v=List()); for(i=0,logint(n,3)+1, my(t=3^i); t<<=if(t>n, 0, exponent(n\t)+1); listput(v,t)); Set(v)[1] \\ Charles R Greathouse IV, Jul 09 2021
  • R
    f <- function(n) nextn(n, factors = c(2,3))
a <- matrix(1:256,ncol=1)
apply(a,1,f)
Showing 1-5 of 5 results.

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

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