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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A360379 a(n) = number of the antidiagonal of the Wythoff array (A035513) that includes prime(n).

Original entry on oeis.org

2, 3, 4, 3, 4, 6, 7, 8, 7, 6, 9, 8, 11, 17, 7, 21, 23, 24, 26, 13, 10, 14, 21, 10, 10, 39, 40, 13, 27, 19, 49, 12, 53, 23, 17, 37, 11, 63, 41, 14, 69, 29, 12, 47, 76, 10, 81, 35, 55, 88, 12, 92, 18, 26, 40, 101, 65, 104, 67, 108, 44, 30, 118, 75, 120, 22
Offset: 1

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Author

Clark Kimberling, Feb 05 2023

Keywords

Comments

Conjecture: Only a finite number of positive integers are missing.

Examples

			The first 6 antidiagonals of the Wythoff array are (1), (2,4), (3,7,6), (5,11,10,9), (8,18,16,15,12), (12,29,26,24,20,14). The 10th prime is 29, which occurs in antidiagonal 6, so a(10) = 6.

Crossrefs

Cf. A000040, A035513, A332938, A360376, A360377, A360378, A360380.

Programs

  • Mathematica
    W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[W[n - k + 1, k], {n, 300}, {k, n, 1, -1}];
Map[#[[1]] &, Most[Reap[NestWhileList[# + 1 &, 1,
Length[Sow[FirstPosition[t, Prime[#]]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 08 2023 *)

A360380 a(n) = number of the diagonal of the Wythoff array, A035513, that includes prime(n). See Comments.

Original entry on oeis.org

1, 2, 3, 0, 1, 5, -6, -7, -4, 3, -6, -3, -8, -16, 4, -20, -22, -23, -25, -8, -3, -9, -18, 9, -1, -38, -39, -6, -24, -14, -48, -3, -52, -18, -10, -34, 0, -62, -38, -5, -68, -24, -1, -44, -75, 7, -80, -30, -52, -87, 11, -91, -9, -19, -35, -100, -62, -103, -64
Offset: 1

Views

Author

Clark Kimberling, Feb 07 2023

Keywords

Comments

The indexing of diagonals is given in A191360. Conjecture: every integer occurs infinitely many times in this sequence; i.e., every diagonal includes infinitely many primes.

Examples

			a(n) = A191360(prime(n)).

Crossrefs

Cf. A000040, A035513, A191360, A332938, A360376, A360377, A360378, A360379.

Programs

  • Mathematica
    w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[w[n - k + 1, k], {n, 300}, {k, n, 1, -1}];
Map[1 + #[[1]] - 2 #[[2]] &, Most[Reap[NestWhile[# + 1 &, 1,
Length[Sow[FirstPosition[t, Prime[#]]]] > 1 &]][[2]][[1]]]]
(* Peter J. C. Moses, Feb 07 2023 *)

A360406 a(n) = minimal positive k such that prime(n) * prime(n+1) * ... * prime(n+k) - 1 is divisible by prime(n+k+1), or -1 if no such k exists.

Original entry on oeis.org

1, 1, 9, 14, 31, 826, 1, 34
Offset: 1

Views

Author

Scott R. Shannon, Feb 06 2023

Keywords

Comments

Assuming a(9) exists it is greater than 1.75 million.
a(11) = 692, a(12) = 8, a(13) = 792. - Robert Israel, Feb 22 2023

Examples

			a(1) = 1 as prime(1) * prime(2) - 1 = 2 * 3 - 1 = 5, which is divisible by prime(3) = 5.
a(2) = 1 as prime(2) * prime(3) - 1 = 3 * 5 - 1 = 14, which is divisible by prime(4) = 7.
a(3) = 9 as prime(3) * ... * prime(12) - 1 = 1236789689134, which is divisible by prime(13) = 41.

Crossrefs

Cf. A360376, A360297, A000040, A007504, A332542, A332580.

Programs

  • Maple
    f:= proc(n) local P,k,p;
P:= ithprime(n); p:= nextprime(P);
for k from 0 to 10^6 do
  if P-1 mod p = 0 then return k fi;
  p:= nextprime(p);
 od;
FAIL
end proc:
map(f, [$1..8]); # Robert Israel, Feb 22 2023
  • Python
    from sympy import prime, nextprime
def A360406(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p*q, 1
    while (s-1)%(q:=nextprime(q)):
        k += 1
        s *= q
    return k # Chai Wah Wu, Feb 06 2023
Showing 1-3 of 3 results.

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