A373631 Least prime p that for some integer k there exist at least n positive primes of the form q!-p for n consecutive integers q belonging to {k,k+1,k+2,..k+n-1}.

3, 7, 11, 17, 191, 50377, 92761, 2085737, 30357947, 1165922581, 8547418259, 35368198867

Offset

1,1

Comments

n=1,{3}!-3.

n=2,{4,5}!-7.

n=3,{4,5,6}!-11.

n=4,{9,10,11,12}!-17.

n=5,{22,23,24,25,26}!-191.

n=6,{12,13,14,15,16,17}!-50377.

n=7,{11,12,13,14,15,16,17}!-92761.

n=8,{10,12,13,14,15,16,17}!-2085737.

n=9,{11,12,13,14,15,16,17,18,19}!-30357947.

n=10,{16,17,18,19,20,21,22,23,24,25}!-1165922581.

n=11,{14,15,16,17,18,19,20,21,22,23,24}!-8547418259.

n=12,{15,16,17,18,19,20,21,22,23,24,25,26}!-35368198867.

Links

Table of n, a(n) for n=1..12.

Mrexcel, Primes with the form q!-p for prime p and n consecutive integers q∈{k,k+1,k+2,..k+n-1}

Example

a(4) = 17 because there exist 4 consecutive integers q {9,10,11,12} such that the numbers q!-17 are all positive primes.

Mathematica

t[n_] :=
 Module[{m = 0, s = {}},
  For[k = 1, k < 20, k++, If[k! > n && PrimeQ[k! - n], m++;
    AppendTo[s, k]; ]]; {Length@s, n, s,
   Minus[Subtract @@@ Partition[s, 2, 1]]}];
a[n_] :=
 If[n == 1, 3, For[r = 1, r <= 10^5, r++, s = Prime@r; v = t[s];
   If[v[[1]] > 0 && SequenceCount[v[[4]], PadLeft[{}, n - 1, 1]] > 0,
    Return[v[[2]]]; ]]]
Table[a[n], {n, 7}]

Crossrefs

Sequence in context: A072456 A138659 A020590 * A063437 A363359 A190711

Adjacent sequences: A373628 A373629 A373630 * A373632 A373633 A373634

Keyword

nonn,more

Author

Zhining Yang, Jun 11 2024

Status

approved