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 1-3 of 3 results.

A063486 a(n) = 2^(2^n) + 5.

Original entry on oeis.org

7, 9, 21, 261, 65541, 4294967301, 18446744073709551621, 340282366920938463463374607431768211461, 115792089237316195423570985008687907853269984665640564039457584007913129639941
Offset: 0

Views

Author

Jason Earls, Jul 28 2001

Keywords

References

  • D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 238.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Example 5.1 on page 153.

Crossrefs

Programs

  • Mathematica
    2^2^Range[0, 10] + 5 (* Paolo Xausa, Apr 17 2024 *)
  • PARI
    for(n=0,8,print1(2^(2^n)+5, ", "))
    
  • PARI
    { for (n=0, 11, write("b063486.txt", n, " ", 2^(2^n) + 5) ) } \\ Harry J. Smith, Aug 23 2009
    
  • SageMath
    [2**2**n + 5 for n in (0..8)] # Stefano Spezia, Jul 20 2025

A130730 Fermat numbers of order 7 or F(n,7) = 2^(2^n)+7.

Original entry on oeis.org

9, 11, 23, 263, 65543, 4294967303, 18446744073709551623, 340282366920938463463374607431768211463, 115792089237316195423570985008687907853269984665640564039457584007913129639943
Offset: 0

Views

Author

Cino Hilliard, Jul 05 2007

Keywords

Comments

This sequence is equivalent to F(n)+ 6 or 2^(2^n)+ 1 + 6. This sequence does not appear to have any special divisibility properties. Fermat numbers of order 5 which are found in A063486, have the divisibility property if n is even, then 7 divides F(n,5). After the first 2 terms the ending digit is the same for all F(n,m) and is (6+m) mod 10.

Crossrefs

Programs

  • Magma
    [2^(2^n)+7: n in [0..11]]; // Vincenzo Librandi, Jan 09 2013
  • Mathematica
    Table[(2^(2^n) + 7), {n, 0, 15}] (* Vincenzo Librandi, Jan 09 2013 *)
  • PARI
    fplusm(n,m)= { local(x,y); for(x=0,n, y=2^(2^x)+m; print1(y",") ) }
    

Formula

F(n,m): The n-th Fermat number of order m = 2^(2^n)+ m. The traditional Fermat numbers are F(n,1) or Fermat numbers of order 1 in this nomenclature.

A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

Original entry on oeis.org

40, 2100, 324000, 117771500, 86640153600, 115851776040000, 260111401804800000, 922852527136155000000, 4931966428685936640000000, 38193820496218904209973280000, 415101787718859995456102400000000
Offset: 3

Views

Author

Jonathan Vos Post, May 25 2008

Keywords

Comments

The array A(k,n) = Product of first n k-gonal pyramidal numbers begins:
===================================================================
..|n=1|n=2|..n=3|...n=4..|......n=5....|......n=6......|......n=7......|.......n=8.........|
k=3|.1.|.4.|..40.|....800.|.......28000.|.......1568000.|.....131712000.|.......15805440000.|A087047
k=4|.1.|.5.|..70.|...2100.|......115500.|......10510500.|....1471470000.|......300179880000.|
k=5|.1.|.6.|.108.|...4320.|......324000.|......40824000.|....8001504000.|.....2304433152000.|
k=6|.1.|.7.|.154.|...7700.|......731500.|.....117771500.|...29678418000.|....11040371496000.|
k=7|.1.|.8.|.208.|..12480.|.....1435200.|.....281299200.|...86640153600.|....39507910041600.|
k=8|.1.|.9.|.270.|.718900.|.....2551500.|.....589396500.|..214540326000.|...115851776040000.|
===================================================================

Examples

			a(3) = product of the first 3 triangular pyramidal (tetrahedral) numbers (A000292) = A087047(3) = 1 * 4 * 10 = 40.
a(4) = product of the first 4 square pyramidal numbers (A000330) = 1 * 5 * 14 * 30 = 2100.
a(5) = product of the first 5 pentagonal pyramidal numbers (A002411) = 1 * 6 * 18 * 40 * 75 = 324000.
a(6) = product of the first 6 hexagonal pyramidal numbers (A002412) = 1 * 7 * 22 * 50 * 95 * 161 = 117771500.
a(7) = product of the first 7 heptagonal pyramidal numbers (A002413) = 1 * 8 * 26 * 60 * 115 * 196 * 308 = 86640153600.
a(8) = product of the first 8 octagonal pyramidal numbers (A002414) = 1 * 9 * 30 * 70 * 135 * 231 * 364 * 540 = 115851776040000.
		

Crossrefs

Programs

  • Maple
    A130729 := proc(n) n!*(n+1)!*(n-2)^n*pochhammer(1+(5-n)/(n-2),n)/6^n ; end: seq(A130729(n),n=3..30) ; # R. J. Mathar, May 31 2008

Formula

A(k,n) = PRODUCT[j=1..n] (1/6)*j*(j+1)*[(k-2)*j+(5-k)].
a(n) ~ Pi^(3/2) * n^(4*n + 1/2) / (2^(n - 3/2) * 3^(n-1) * exp(3*n+2)) * (1 + (3*log(n) + 3*gamma + 5/4)/n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 29 2023

Extensions

More terms from R. J. Mathar, May 31 2008
Showing 1-3 of 3 results.