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.

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A352537 Primes whose position in the Wythoff array is immediately followed by a prime both in the next column and the next row.

Original entry on oeis.org

2, 3, 919, 1223, 1699, 3329, 8009, 11717, 13691, 19079, 20921, 21011, 22643, 22739, 24623, 26309, 28571, 28619, 28979, 30389, 33629, 34739, 35257, 41179, 42577, 48647, 54133, 58601, 59627, 61511, 65171, 70979, 75707, 80141, 84221, 86869, 90677, 93557, 94781
Offset: 1

Views

Author

Michel Marcus, Mar 20 2022

Keywords

Examples

			The Wythoff array begins:
   1    2    3    5   ...
   4    7   11   18   ...
   6   10   16   26   ...
   ...
where one can see these 2 patterns:
   2    3   and   3    5
   7             11
so 2 and 3 are terms.

Crossrefs

Cf. A003603, A035612, A035513 (Wythoff array).
Intersection of A352538 and A352539.

Programs

  • PARI
    T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break);););
isokp(m) = my(pos = cell(prime(m))); isprime (T(pos[1], pos[2]+1)) && isprime(T(pos[1]+1, pos[2]));
lista(nn) = for (n=1, nn, if (isokp(n), print1(prime(n), ", ")));

A352539 Primes whose position in the Wythoff array is immediately followed by another prime in the next row.

Original entry on oeis.org

2, 3, 13, 17, 59, 71, 101, 157, 347, 359, 401, 683, 821, 881, 919, 1063, 1223, 1613, 1699, 1787, 1931, 2081, 2333, 2663, 2711, 2909, 2999, 3011, 3299, 3329, 3371, 3389, 3623, 3821, 3911, 4019, 4049, 4337, 4349, 4481, 4931, 5171, 5273, 5651, 5741, 5849, 5879, 6029, 6079
Offset: 1

Views

Author

Michel Marcus, Mar 20 2022

Keywords

Examples

			The Wythoff array begins:
   1    2    3    5    8   13  ...
   4    7   11   18   29   47  ...
   6   10   16   26   42   68  ...
   ...
So 2, 3 and 13 are terms since they are vertically followed by 7, 11 and 47.

Crossrefs

Cf. A003603, A035612, A035513 (Wythoff array).
Cf. A352537 (next row and column), A352538 (next column).

Programs

  • PARI
    T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break);););
isokv(m) = my(pos = cell(prime(m))); isprime (T(pos[1]+1, pos[2]));
lista(nn) = for (n=1, nn, if (isokv(n), print1(prime(n), ", ")));

A352583 a(n) is the value of the cell in the Wythoff array that lies in the next row and same column as the cell containing n.

Original entry on oeis.org

4, 7, 11, 6, 18, 9, 10, 29, 12, 15, 16, 14, 47, 17, 20, 24, 19, 26, 22, 23, 76, 25, 28, 32, 27, 39, 30, 31, 42, 33, 36, 37, 35, 123, 38, 41, 45, 40, 52, 43, 44, 63, 46, 49, 50, 48, 68, 51, 54, 58, 53, 60, 56, 57, 199, 59, 62, 66, 61, 73, 64, 65, 84, 67, 70, 71, 69, 102, 72, 75
Offset: 1

Views

Author

Michel Marcus, Mar 21 2022

Keywords

Comments

From Kevin Ryde, Jun 05 2022: (Start)
a(n) is n with the "odd" part (A348853) of its Zeckendorf representation increased to the next greater "odd" number.
This increase is Zeckendorf digits +10 or +100 at the odd part, according to whether the final digits there are ..101 or ..001, respectively.
A354321(n) is the first of those three digits so that a(n) = n + Fibonacci(A035612(n) + 3 - A354321(n)).
(End)

Examples

			The Wythoff array (A035513 or A083412) begins:
   1    2    3    5    8 ...
   4    7   11   18   29 ...
   6   10   16   26   42 ...
   ...
so a(1) = 4, a(2) = 7, a(3) = 11, a(4) = 6, ...

Crossrefs

Cf. A035513 and A083412 (Wythoff array), A003603 (row number), A035612 (column number).
Cf. A348853 (odd part), A354321 (above 01), A000045 (Fibonacci numbers).
Cf. A022342 (same row, next column).
Cf. A349102 (binary increase odd).

Programs

  • PARI
    T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break););); \\ see A003603 and A035612
a(n) = {my(pos = cell(n)); T(pos[1]+1, pos[2]);}
  • PARI
    { my(phi=quadgen(5),s=phi-1,c=2*phi-3);
a(n) = my(t=n,k=3,r);
  until(r
Previous Showing 11-13 of 13 results.

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