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.

A004231 Ackermann's sequence: n^^n := n^n^n^...^n (with n n's).

Original entry on oeis.org

1, 1, 4, 7625597484987
Offset: 0

Views

Author

Daniel Wild (wild(AT)edumath.u-strasbg.fr)

Keywords

Comments

Using Knuth's arrow notation, this is n^^^2 (n-penta-2) or n^^n (n-tetra-n). - Andrew Robbins, Apr 16 2009
Comment from Trevor Green: The fourth term in this sequence has about as many digits - 8.07 * 10^153 - as the *square* of the number of protons in the universe.
We could prepend a(0) = 1 (since 0^^0 = 1, that is, the "empty power tower" gives the "empty product"). - Daniel Forgues, May 17 2013
The last 60 decimal digits of a(4) are ...67586985427238232605843019607448189676936860456095261392896. - Daniel Forgues, Jun 25 2016
From Daniel Forgues, Jul 06 2016: (Start)
a(4) has (the following number having 154 decimal digits)
80723047260282253793826303970853990300713679217387 \
43031867082828418414481568309149198911814701229483 \
451981557574771156496457238535299087481244990261351117 decimal digits.
a(4) = 4^4^4^4 = 4^
13407807929942597099574024998205846127479365820592 \
39337772356144372176403007354697680187429816690342 \
7690031858186486050853753882811946569946433649006084096,
the exponent of 4 having 155 decimal digits. (End)
The fractional part of 4^4^4*log[10](4) starts .373100157363599870..., so the first few digits of a(4) are 23610226714597313.... - Robert Israel, Jul 06 2016

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 60.

Links

Crossrefs

Cf. A027747, A008868 (which mentions an older estimate for the same number), A266200.
Main diagonal of A321312.

Programs

  • Maple
    b:= (n, i)-> `if`(i=0, 1, n^b(n, i-1)):
a:= n-> b(n, n):
seq(a(n), n=0..3);  # Alois P. Heinz, Aug 22 2017
  • Mathematica
    a[n_] := If[n == 0, 1, Nest[n^#&, n, n-1]];
Table[a[n], {n, 0, 3}] (* Jean-François Alcover, Mar 19 2019 *)

A321312 A(n,k) = n^^k is the k-th tetration of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 4, 3, 1, 0, 1, 16, 27, 4, 1, 1, 1, 65536, 7625597484987, 256, 5, 1
Offset: 0

Views

Author

Alois P. Heinz, Nov 03 2018

Keywords

Examples

			Square array A(n,k) begins:
  1, 0,      1,              0,      1,   0,   1, ...
  1, 1,      1,              1,      1,   1,   1, ...
  1, 2,      4,             16,  65536, ...
  1, 3,     27,  7625597484987,    ...
  1, 4,    256,            ...
  1, 5,   3125,            ...
  1, 6,  46656,            ...
  1, 7, 823543,            ...
  ...

Links

Crossrefs

Columns k=0-3 give: A000012, A001477, A000312, A002488.
Rows n=0-4 give: A059841, A000012, A014221, A014222(k+1), A114561(k+1).
Main diagonal gives A004231 (Ackermann's sequence).
Cf. A027747, A171882 (by upwards diagonals).

Programs

  • Maple
    A:= (n, k)-> `if`(k=0, 1, n^A(n, k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..6);

A265110 Partial row products of table A027746, prime factors with repetition.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 2, 6, 7, 2, 4, 8, 3, 9, 2, 10, 11, 2, 4, 12, 13, 2, 14, 3, 15, 2, 4, 8, 16, 17, 2, 6, 18, 19, 2, 4, 20, 3, 21, 2, 22, 23, 2, 4, 8, 24, 5, 25, 2, 26, 3, 9, 27, 2, 4, 28, 29, 2, 6, 30, 31, 2, 4, 8, 16, 32, 3, 33, 2, 34, 5, 35, 2, 4, 12, 36
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 01 2015

Keywords

Comments

T(n,1) = A020639(n); T(n,A001222(n)) = n.

Examples

			.   n |   T(n,*)       |   A027746(n,*)
. ----+----------------+----------------
.   1 |   1            |   1
.   2 |   2            |   2
.   3 |   3            |   3
.   4 |   2, 4         |   2, 2
.   5 |   5            |   5
.   6 |   2, 6         |   2, 3
.   7 |   7            |   7
.   8 |   2, 4, 8      |   2, 2, 2
.   9 |   3, 9         |   3, 3
.  10 |   2, 10        |   2, 5
.  11 |   11           |   11
.  12 |   2, 4, 12     |   2, 2, 3
.  13 |   13           |   13
.  14 |   2, 14        |   2, 7
.  15 |   3, 15        |   3, 5
.  16 |   2, 4, 8, 16  |   2, 2, 2, 2
.  17 |   17           |   17
.  18 |   2, 6, 18     |   2, 3, 3
.  19 |   19           |   19
.  20 |   2, 4, 20     |   2, 2, 5

Links

Crossrefs

Cf. A027746, A175943, A001222 (row lengths), A020639.

Programs

  • Haskell
    a265110 n k = a265110_tabf !! (n-1) !! (k-1)
a265110_row n = a265110_tabf !! (n-1)
a265110_tabf = map (scanl1 (*)) a027746_tabf
  • Mathematica
    Table[FoldList[Times, Flatten[FactorInteger[n] /. {p_, e_} /; e > 0 :> ConstantArray[p, e]]], {n, 37}] // Flatten (* Michael De Vlieger, Apr 28 2017 *)

Formula

T(n,k) = product(A027747(n,k): k = 1 .. A001221(n)).

A094894 a(n) = prime(Lucas(n)), Lucas numbers beginning at 2 (A000032).

Original entry on oeis.org

3, 2, 5, 7, 17, 31, 61, 109, 211, 383, 677, 1217, 2137, 3733, 6521, 11279, 19463, 33347, 56963, 97159, 165443, 280549, 474809, 801611, 1351841, 2273989, 3821689, 6412541, 10742339, 17974841, 30045019, 50163697, 83669419, 139420003, 232106309, 386086573, 641716373
Offset: 0

Views

Author

N. J. A. Sloane, Jun 15 2004

Keywords

Links

Crossrefs

Cf. A027747, A030427, A000032, A000045, A000040.

Programs

  • Magma
    [NthPrime(Lucas(n)): n in [0..40]]; // Vincenzo Librandi, Jan 14 2016
  • Mathematica
    Table[ Prime[ Fibonacci[n + 1] + Fibonacci[n - 1]], {n, 0, 35}] (* Robert G. Wilson v, Jun 16 2004 *)
Table[Prime[LucasL[n]], {n, 0, 40}] (* Vincenzo Librandi, Jan 14 2016 *)

Formula

a(n) = prime(L(n)) = A000040(A000032(n)) = prime(Fib(n + 1) + Fib(n-1)), where Fib are the Fibonacci numbers(A000045) with Fib(-1) = -1.

Extensions

More terms from Robert G. Wilson v, Jun 16 2004
a(34)-a(36) from Vincenzo Librandi, Jan 14 2016

A051441 4^3^2^1, 5^4^3, 6^5, 7.

Original entry on oeis.org

262144, 542101086242752217003726400434970855712890625, 7776, 7
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A027747.
Showing 1-5 of 5 results.

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

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