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A006278 Numbers that are a product of successive primes congruent to 1 (mod 4). 9
5, 65, 1105, 32045, 1185665, 48612265, 2576450045, 157163452745, 11472932050385, 1021090952484265, 99045822390973705, 10003628061488344205, 1090395458702229518345, 123214686833351935572985 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n)+2 is prime for n=0,1. No others are prime for n <= 200. Compare A002110 and A078586. - T. D. Noe, Dec 01 2002

Also, a(n) is least hypotenuse of exactly A003462(n+1) Pythagorean triangles of which 2^n are primitive. - Lekraj Beedassy, Dec 06 2003

Also, a(n) are the record setting values of m, for the number of solutions to "m*k-1 is a square", for some k, 1<=k<m. There is one solution for m=2, and for a given m = a(n) there are 2^n solutions. For a given m there also 2^(n-1) solutions for primitively representing m as x^2 + y^2. See A008782. Also compare with A102476, which applies to "m*k+1 is a square". - Richard R. Forberg, Mar 18 2016

LINKS

T. D. Noe, Table of n, a(n) for n=1..100

Eric Weisstein's World of Mathematics, Pythagorean Triple.

MATHEMATICA

maxN=15; pLst={}; k=0; While[Length[pLst]<maxN, k++; If[PrimeQ[4k+1], AppendTo[pLst, 4k+1]]]; lst=Drop[FoldList[Times, 1, pLst], 1]

Rest[FoldList[Times, 1, Select[Prime[Range[50]], Mod[#, 4]==1&]]] (* Harvey P. Dale, Jun 16 2013 *)

result = {}; Do[count = 0;

Do[If[IntegerQ[Sqrt[m*k - 1]], count++, {k, 1, m - 1}]; If[count > 0, AppendTo[result, {m, count}]], {m, 2, 1105}]; result  (* Richard R. Forberg, Mar 18 2016 *)

CROSSREFS

Cf. A002110, A078586, A185952.

Sequence in context: A208588 A249930 A207262 * A234871 A121822 A056245

Adjacent sequences:  A006275 A006276 A006277 * A006279 A006280 A006281

KEYWORD

nonn

AUTHOR

Gene_Salamin(AT)cohr.com

STATUS

approved

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Last modified May 25 15:52 EDT 2017. Contains 287039 sequences.