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

%I A006278 #58 Jun 22 2025 07:11:52
%S A006278 5,65,1105,32045,1185665,48612265,2576450045,157163452745,
%T A006278 11472932050385,1021090952484265,99045822390973705,
%U A006278 10003628061488344205,1090395458702229518345,123214686833351935572985
%N A006278 a(n) is the product of the first n primes congruent to 1 (mod 4).
%C A006278 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
%C A006278 Also, a(n) is least hypotenuse of exactly A003462(n) Pythagorean triangles of which 2^(n-1) are primitive. - _Lekraj Beedassy_, Dec 06 2003
%C A006278 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
%C A006278 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
%H A006278 T. D. Noe, <a href="/A006278/b006278.txt">Table of n, a(n) for n = 1..100</a>
%H A006278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A006278 a(n) = Product_{i=1..n} A002144(i). - _Alois P. Heinz_, Mar 01 2021
%t A006278 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]
%t A006278 Rest[FoldList[Times,1,Select[Prime[Range[50]],Mod[#,4]==1&]]] (* _Harvey P. Dale_, Jun 16 2013 *)
%o A006278 (PARI) tree(v)=my(t=#v); if(t<4, factorback(v), tree(v[1..t\2])*tree(v[t\2+1..t]));
%o A006278 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
%Y A006278 Cf. A002110, A002144, A038346, A078586, A185952.
%K A006278 nonn
%O A006278 1,1
%A A006278 Gene_Salamin(AT)cohr.com