A336280 Number of consecutive primes of the form k*prime(n) + 1, starting with the least such prime A035095(n), that divides the Wendt determinant A048954(prime(n)).

1, 1, 2, 1, 4, 3, 2, 3, 2, 3, 1, 5, 6, 2, 6, 3, 3, 3, 1, 6, 3, 5, 5, 7, 5, 5, 6, 7, 4, 7, 5, 10, 10, 4, 4, 6, 10, 3, 4, 12, 11, 5, 7, 8, 7, 8, 11, 4, 4, 4, 14, 7, 11, 7, 13, 11, 13, 7, 18, 18, 6, 7, 17, 12, 9, 7, 9, 14, 12, 9, 16, 14, 11, 13, 10

Offset: 1

Author

Frank M Jackson and Michael B Rees, Jul 15 2020

Keywords

• nonn

Comments

Michael B Rees has conjectured that:

1. for every prime p, the Wendt determinant Wendt(p) has all its prime factors that are greater than p of the form k*p + 1.

2. for every prime p = prime(n) and its corresponding Wendt determinant W(p) there exists a finite number of m consecutive primes (p_1,p_2,..,p_m) of the form k*p + 1 that will divide Wendt(p) where p_1 is always the least prime of the form k*p + 1.

This sequence gives the value m for each p = prime(n).

Examples

			a(6) = 3 gives p = prime(6) = 13 and W(13) = 3^6*53^2*79^2*131^2*521^2*8191. The sequence of primes of the form q = k*13 + 1, starting with the least such prime 53 that divide W(11) is (53, 79, 131). The sequence has 3 terms.
		

Links

• Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114).
• Eric Weisstein's World of Mathematics, Circulant matrix .
• Wikipedia, Circulant matrix .

Crossrefs

Cf. A035095, A048954.

Programs

Mathematica

w[n_] := Module[{x}, Resultant[x^n-1, (1+x)^n-1, x]]; k[n_, m_] := Module[{p=Prime@n, q=0, lst={}}, Do[q++; While[! PrimeQ[p*q+1], q++]; AppendTo[lst, q], {m}]; lst];
lst1 = {}; Do[lst=k[n, 50]*Prime[n]+1; m = 1; Do[If[IntegerQ[w[Prime[n]]/lst[[m]]]&&m<=Length@lst, m++, Break[]], {Length@lst}]; AppendTo[lst1, m-1], {n, 1, 75}]; lst1