

A006278


Numbers that are a product of successive primes congruent to 1 (mod 4).


12



5, 65, 1105, 32045, 1185665, 48612265, 2576450045, 157163452745, 11472932050385, 1021090952484265, 99045822390973705, 10003628061488344205, 1090395458702229518345, 123214686833351935572985
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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^(n1) are primitive.  Lekraj Beedassy, Dec 06 2003
Also, a(n) are the record setting values of m, for the number of solutions to "m*k1 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^(n1) 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


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

Cf. A002110, A038346, 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



