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A006278
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a(n) is the product of the first n primes congruent to 1 (mod 4).
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16
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5, 65, 1105, 32045, 1185665, 48612265, 2576450045, 157163452745, 11472932050385, 1021090952484265, 99045822390973705, 10003628061488344205, 1090395458702229518345, 123214686833351935572985
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Gene_Salamin(AT)cohr.com
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STATUS
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approved
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