A006278 a(n) is the product of the first n primes congruent to 1 (mod 4).

5, 65, 1105, 32045, 1185665, 48612265, 2576450045, 157163452745, 11472932050385, 1021090952484265, 99045822390973705, 10003628061488344205, 1090395458702229518345, 123214686833351935572985

Offset: 1

Author

Gene_Salamin(AT)cohr.com

Keywords

• nonn

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

Crossrefs

Cf. A002110, A002144, A038346, A078586, A185952.

Programs

Mathematica

maxN=15; pLst={}; k=0; While[Length[pLst]Harvey P. Dale, Jun 16 2013 *)

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(#PCharles R Greathouse IV, Jan 08 2018

Formula

a(n) = Product_{i=1..n} A002144(i). - Alois P. Heinz, Mar 01 2021