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 1-9 of 9 results.

A199582 Central terms of the triangle in A199333: a(n) = A199333(n,[n/2]).

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 29, 53, 107, 211, 431, 809, 1619, 3037, 6079, 11467, 22937, 43541, 87083, 166183, 332393, 636761, 1273541, 2448049, 4896103, 9438851, 18877711, 36484271, 72968563, 141332173, 282664351, 548544487, 1097088989, 2132671027, 4265342057
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

a(n) = max(A199333(n,k): 0<=k<=n/2) = A006530(A199695(n)) = A006530(A199696(n));
a(2*n) = A007918(a(2*n-1)) for n > 0.

Links

Crossrefs

Cf. A199581, A001405.

Programs

  • Haskell
    a199582 n = a199333_row n !! (n `div` 2)

A199695 Row products of the triangle in A199333.

Original entry on oeis.org

1, 1, 2, 9, 175, 8281, 1856261, 649893049, 817291210163, 1847322434679121, 14368726069959027071, 342031303262647675287601, 13964481217238950868653586531, 1889891784470148590323094656731121, 586215019967842464352819482405063771511
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

a(n) = Product_{k=0..n} A199333(n,k);
A199696(n) = A007947(a(n));
A020639(a(n)) = A008578(n); A006530(a(n)) = A199582(n).

Crossrefs

Cf. A199333, A199694.

Programs

  • Haskell
    a199695 = product . a199333_row

A199425 Number of distinct primes in rows 0 through n of the triangle in A199333.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 30, 35, 42, 49, 57, 63, 71, 80, 90, 99, 110, 121, 133, 145, 157, 170, 184, 197, 212, 227, 242, 258, 275, 292, 310, 327, 345, 364, 384, 404, 425, 446, 467, 489, 512, 535, 558, 581, 606, 630, 656, 682, 709, 736, 764
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Links

Crossrefs

Cf. A199424, A126256.

Programs

  • Haskell
    import Data.List (union)
a199425 n = a199425_list !! n
a199425_list = f [] a199333_tabl where
   f ps (ts:tts) =  (length ps') : f ps' tts where
     ps' = ps `union` (take ((length ts - 1) `div` 2) $ tail ts)

A199696 Products of distinct terms in n-th row of the triangle in A199333.

Original entry on oeis.org

1, 1, 2, 3, 35, 91, 7337, 25493, 9351479, 42980489, 78695113801, 584834423801, 4754839123536133, 43472885623916761, 1887750276489057845213, 21019416307292530253881, 4675204650607654300508731931, 77008997457626136207428248409
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

a(n) = Product_{k=0..floor(n/2)} A199333(n,k);
A020639(a(n)) = A008578(n);
A006530(a(n)) = A199582(n).

Programs

  • Haskell
    a199696 n = product . (take (n `div` 2 + 1)) $ a199333_row n

Formula

a(n) = A007947(A199695(n)).

A199424 Index of first row in triangle A199333 containing n-th prime.

Original entry on oeis.org

2, 3, 4, 4, 6, 5, 8, 9, 6, 6, 12, 7, 14, 15, 16, 7, 18, 19, 20, 9, 22, 23, 24, 25, 8, 27, 28, 8, 30, 31, 11, 33, 34, 35, 36, 9, 12, 39, 40, 41, 42, 43, 13, 45, 46, 47, 9, 10, 50, 14, 52, 53, 54, 55, 56, 57, 58, 15, 60, 61, 62, 63, 64, 65, 66, 16, 11, 69, 70
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

a(n) <= n + 1.

Links

Crossrefs

Cf. A199425, A058084.

Programs

  • Haskell
    import Data.List (findIndex)
import Data.Maybe (fromJust)
a199424 n = fromJust $ findIndex (elem $ a000040 n) a199333_tabl

A199581 Central terms of the triangle in A199333: a(n) = A199333(2*n,n).

Original entry on oeis.org

1, 2, 7, 29, 107, 431, 1619, 6079, 22937, 87083, 332393, 1273541, 4896103, 18877711, 72968563, 282664351, 1097088989, 4265342057, 16608401041, 64758466127, 252814859149, 988089813541, 3865761355523, 15138431958437, 59333638261529, 232737382916429
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

a(n) = A199582(2*n).

Links

Crossrefs

Cf. A000984, A199333, A199582.

Programs

  • Haskell
    a199581 n = a199333_row (2*n) !! n

A199694 Row sums of the triangle in A199333.

Original entry on oeis.org

1, 2, 4, 8, 19, 42, 99, 208, 443, 906, 1853, 3780, 7595, 15246, 30537, 61160, 122413, 244930, 489985, 980080, 1960257, 3920646, 7841419, 15682972, 31366187, 62732582, 125465509, 250931320, 501863047, 1003726454, 2007453193, 4014906880, 8029814297
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

a(n) = sum(A199333(n,k): 0 <= k <= n).

Links

Crossrefs

Cf. A199695.

Programs

  • Haskell
    a199694 = sum . a199333_row

A362034 Triangle read by rows: T(n,0) = T(n,n) = 2, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 2, 7, 7, 2, 2, 11, 17, 11, 2, 2, 13, 29, 29, 13, 2, 2, 17, 43, 59, 43, 17, 2, 2, 19, 61, 103, 103, 61, 19, 2, 2, 23, 83, 167, 211, 167, 83, 23, 2, 2, 29, 107, 251, 379, 379, 251, 107, 29, 2, 2, 31, 137, 359, 631, 761, 631, 359, 137, 31, 2
Offset: 0

Views

Author

Jack Braxton, Apr 05 2023

Keywords

Comments

In order to get the next number in the row, you add the two numbers above it, and find the next prime.
3 is the only prime number that never shows up.
5 is the only prime number that only shows up once; every prime number above 5 shows up at least twice.

Examples

			Triangle begins:
      k=0  1   2   3   4   5   6   7   8  9 10
  n=0:  2
  n=1:  2  2
  n=2:  2  5   2
  n=3:  2  7   7   2
  n=4:  2 11  17  11   2
  n=5:  2 13  29  29  13   2
  n=6:  2 17  43  59  43  17   2
  n=7:  2 19  61 103 103  61  19   2
  n=8:  2 23  83 167 211 167  83  23   2
  n=9:  2 29 107 251 379 379 251 107  29  2
 n=10:  2 31 137 359 631 761 631 359 137 31  2

Links

Crossrefs

Cf. A000040, A007318, A132403, A199333.

Programs

  • Maple
    for n from 0 to 10 do
  T[n,0]:= 2: T[n,n]:= 2:
  for k from 1 to n-1 do
    T[n,k]:= nextprime(T[n-1,k-1]+T[n-1,k]-1)
  od
od:
for n from 0 to 10 do
  seq(T[n,k],k=0..n)
od; # Robert Israel, Apr 05 2023
  • Mathematica
    T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2023, after Maple *)
  • PARI
    T(n,k) = if (n==0, 2, if (k==0, 2, if (k==n, 2, nextprime(T(n-1,k-1) + T(n-1,k))))); \\ Michel Marcus, Apr 07 2023

Formula

T(n,k) = A007918(T(n-1,k-1) + T(n-1,k)) for 0 < k < n. - Robert Israel, Apr 05 2023

A132403 Triangle read by rows: T(n,k) = nextprime( T(n-1,k) + T(n-1,k-1) ), where nextprime = A151800.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 5, 11, 11, 5, 7, 17, 23, 17, 7, 11, 29, 41, 41, 29, 11, 13, 41, 71, 83, 71, 41, 13, 17, 59, 113, 157, 157, 113, 59, 17, 19, 79, 173, 271, 317, 271, 173, 79, 19, 23, 101, 257, 449, 593, 593, 449, 257, 101, 23, 29, 127, 359, 709, 1049, 1187, 1049, 709, 359, 127, 29
Offset: 0

Views

Author

Jonathan Vos Post, Nov 12 2007

Keywords

Comments

Each number is the smallest prime > the sum of the 2 numbers above (consider each line padded with 0 on each side).

Examples

			Triangle begins:
  1
  2....2
  3....5....3
  5...11...11....5
  7...17...23...17....7
  11..29...41...41...29...11
  13..41...71...83...71...41...13
  17..59..113..157..157..113...59...17
  19..79..173..271..317..271..173...79...19
  23.101..257..449..593..593..449..257..101...23
  29.127..359..709.1049.1187.1049..709..359..127..29
  31.157..487.1069.1759.2237.2237.1759.1069..487.157..31
  37.191..647.1559.2833.4001.4583.4001.2833.1559.647.191.37
  ...
First column is A008578.
Second column is A064337.

Crossrefs

Cf. A008578, A151800, A007318, A064337, A199333, A362034.
Showing 1-9 of 9 results.

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

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