A336688 Primes p such that the Wendt determinant A048954(p) has prime factors less than p.

3, 7, 13, 31, 73, 127, 307, 331, 757

Offset

1,1

Comments

Michael B Rees has conjectured that for all primes p, each fully exponentiated prime factor less than p that divides the Wendt determinant W(p), if it exists, is of the form k*p + 1.

This sequence identifies the prime index p of Wendt determinants W(p) that have prime factors less than p.

These prime indices appear to be a subset of the lucky primes A031157.

Links

Table of n, a(n) for n=1..9.

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.

Example

a(3) = 13. The Wendt determinant with a prime index p = 13 has prime factors less than p. W(13) = 3^6*53^2*79^2*131^2*521^2*8191 and 3^6 = 729 is of the form k*13 + 1. It is the 3rd occurrence of such a determinant.

Mathematica

w[n_] := Resultant[x^n-1, (1+x)^n-1, x]; getp[n_] := Module[{W=w[n], lst=Table[Prime[m], {m, 1, PrimePi[n]}], lst1={}, j, k, l}, Do[j=1; While[W>0&&IntegerQ[W/lst[[l]]^j], j++]; If[j-1>0, AppendTo[lst1, {lst[[l]], j-1}]], {l, 1, Length@lst}]; Join[{n}, lst1]]; lst = {}; Do[lst1=getp[Prime[n]]; If[Length@lst1>1, AppendTo[lst, lst1[[1]]]], {n, 1, PrimePi[331]}]; lst

Crossrefs

Cf. A048954, A336280.

Sequence in context: A110436 A126879 A247895 * A155128 A176589 A077314

Adjacent sequences: A336685 A336686 A336687 * A336689 A336690 A336691

Keyword

nonn,more

Author

Frank M Jackson and Michael B Rees, Jul 31 2020

Extensions

a(9) from Jinyuan Wang, Sep 04 2020

Status

approved