OFFSET
1,1
COMMENTS
a(n)+2 is prime for n=1,2. 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) Pythagorean triangles of which 2^(n-1) 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
a(n) is the smallest m such that A000089(m) = 2^n. Also, numbers k for which A000089(k) sets a new record. - Jianing Song, Apr 27 2019
LINKS
T. D. Noe, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Pythagorean Triple
FORMULA
a(n) = Product_{i=1..n} A002144(i). - Alois P. Heinz, Mar 01 2021
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 *)
PROG
(PARI) tree(v)=my(t=#v); if(t<4, factorback(v), tree(v[1..t\2])*tree(v[t\2+1..t]));
a(n, x=9*n\4+2)=my(P=select(p->p%4==1, primes(x))); if(#P<n, a(n, 3*x\2+1), tree(P[1..n])) \\ Charles R Greathouse IV, Jan 08 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gene_Salamin(AT)cohr.com
STATUS
approved