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A336688 Primes p such that the Wendt determinant A048954(p) has prime factors less than p. 0
3, 7, 13, 31, 73, 127, 307, 331, 757 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A110436 A126879 A247895 * A155128 A176589 A077314
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

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Last modified June 14 04:14 EDT 2024. Contains 373393 sequences. (Running on oeis4.)