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.

A362035 Numbers that occur three or more times in A362034.

Original entry on oeis.org

2, 17, 29, 43, 59, 61, 83, 103, 107, 137, 167, 173, 211, 251, 257, 307, 359, 379, 419, 479, 499, 541, 613, 631, 673, 691, 761, 769, 853, 887, 937, 991, 1031, 1129, 1151, 1231, 1361, 1399, 1459, 1471, 1493, 1583, 1697, 1823, 1831, 1973, 2111, 2179, 2243, 2251, 2393
Offset: 1

Views

Author

Jack Braxton, Apr 05 2023

Keywords

Comments

All terms are prime because all terms in A362034 are.

Examples

			2 is a term because in A362034, 2 occurs at positions 1, 2, and 3.
17 is a term because in A362034, 17 occurs at positions 13, 23, and 27.

Crossrefs

Cf. A000040, A362034.

Programs

  • Mathematica
    nn = 300; q = Prime[nn]; c[] = 0; p[] = False; T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; TakeWhile[Sort@ Reap[Do[(c[#]++; If[And[c[#] >= 3, ! p[#]], Sow[#]; p[#] = True]) &@ T[n, k], {n, 0, nn}, {k, 0, n}] ][[-1, -1]], # <= q &] (* Michael De Vlieger, Apr 06 2023 *)

A362036 The prime indices of A362034.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 7, 5, 1, 1, 6, 10, 10, 6, 1, 1, 7, 14, 17, 14, 7, 1, 1, 8, 18, 27, 27, 18, 8, 1, 1, 9, 23, 39, 47, 39, 23, 9, 1, 1, 10, 28, 54, 75, 75, 54, 28, 10, 1, 1, 11, 33, 72, 115, 135, 115, 72, 33, 11, 1, 1, 12, 40, 95, 167, 222, 222, 167, 95, 40, 12, 1
Offset: 0

Views

Author

Jack Braxton, Apr 05 2023

Keywords

Examples

			Triangle begins:
      k=0  1  2  3  4
  n=0:  1;
  n=1:  1, 1;
  n=2:  1, 3, 1;
  n=3:  1, 4, 4, 1;
  n=4:  1, 5, 7, 5, 1;
  n=5:  ...

Crossrefs

Cf. A000720, A362034.

Programs

  • 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[PrimePi@ T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2023 *)
  • 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))))); \\ A362034
T(n,k) = primepi(t(n,k)); \\ Michel Marcus, Apr 07 2023

Formula

T(n,k) = A000720(A362034(n,k)).

A362037 Sums of the rows in A362034.

Original entry on oeis.org

2, 4, 9, 18, 43, 88, 183, 370, 761, 1536, 3081, 6202, 12427, 24908, 49875, 99834, 199769, 399572, 799289, 1598726, 3197557, 6395400, 12791045, 25582310, 51164767, 102329800, 204659929, 409320270, 818640731, 1637281876, 3274564199, 6549128694, 13098257783, 26196515972
Offset: 1

Views

Author

Jack Braxton, Apr 05 2023

Keywords

Examples

			a(3) = 9, the 3rd row in A362034 is 2 5 2; and 2 + 5 + 2 = 9.
a(4) = 18, the 4th row in A362034 is 2, 7, 7, 2; and 2+7+7+2 = 18.

Crossrefs

Cf. A362034.

Programs

  • 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]; Total /@ Table[T[n, k], {n, 0, 33}, {k, 0, n}] (* Michael De Vlieger, Apr 06 2023 *)

A199333 Triangle read by rows: T(n,0) = T(n,n) = 1, 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

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 5, 1, 1, 7, 13, 13, 7, 1, 1, 11, 23, 29, 23, 11, 1, 1, 13, 37, 53, 53, 37, 13, 1, 1, 17, 53, 97, 107, 97, 53, 17, 1, 1, 19, 71, 151, 211, 211, 151, 71, 19, 1, 1, 23, 97, 223, 367, 431, 367, 223, 97, 23, 1, 1, 29, 127
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 09 2011

Keywords

Comments

T(n,k) = T(n,n-k);
T(n,0) = 1, cf. A000012;
T(n,1) = A008578(n), n > 0;
A199424(n) = first row in triangle A199302 containing n-th prime;
A199425(n) = number of distinct primes in rows 0 through n;
large terms in the b-file are probable primes only, row number > 50.

Examples

			0:                 1
1:               1   1
2:             1   2   1
3:           1   3   3   1
4:         1   5   7   5   1
5:       1   7  13  13   7   1
6:     1  11  23  29  23  11   1
7:   1  13  37  53  53  37  13   1
8: 1  17  53  97 107  97  53  17   1
primes in 8th row:
T(7,0) + T(7,1) = 1+13 = 14 --> T(8,1) = T(8,7) = 19;
T(7,1) + T(7,2) = 13+37 = 50 --> T(8,2) = T(8,6) = 53, already in row 7;
T(7,2) + T(7,3) = 37+53 = 90 --> T(8,3) = T(8,5) = 97;
T(7,3) + T(7,4) = 53+53 = 106 --> T(8,4) = 107.

Links

Crossrefs

Cf. A159477; A199581 & A199582 (central terms), A199694 (row sums), A199695 & A199696 (row products); A007318.
Cf. A132403, A362034.

Programs

  • Haskell
    a199333 n k = a199333_tabl !! n !! k
a199333_row n = a199333_tabl !! n
a199333_list = concat a199333_tabl
a199333_tabl = iterate
   (\row -> map a159477 $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
  • Mathematica
    T[n_, k_] := T[n, k] = Switch[k, 0|n, 1, _, With[{m = T[n-1, k] + T[n-1, k-1]}, If[PrimeQ[m], m, NextPrime[m]]]];
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

Formula

T(n,k) = A007918(T(n-1,k) + T(n-1,k-1)), 0 < k < n, T(n,0) = T(n,n) = 1.

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 1-5 of 5 results.

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