|
%I #13 Sep 28 2019 21:16:05
%S 2,3,7,23,29,257,1439,2351,26561,146639,1891949,2062889,341708489,
%T 2062232987
%N a(n) is the largest number in an n-term AP of Chen primes.
%C Zhou proves that a(n) exists for each n, generalizing Green & Tao (2008) from primes to Chen primes and generalizing Green & Tao (2006) from 3-AP to n-AP. Sequence is increasing by definition.
%H B. Green and T. Tao, <a href="http://dx.doi.org/10.5802/jtnb.538">Restriction theory of the Selberg sieve, with applications</a>, J. Théor. Nombres Bordeaux 18:1 (2006), pp. 147-182. <a href="http://arXiv.org/abs/math/0405581">arXiv:math/0405581</a> [math.NT]
%H Ben Green and Terence Tao, <a href="http://arXiv.org/abs/math/0404188">The primes contain arbitrarily long arithmetic progressions</a>, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
%H Binbin Zhou, <a href="https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/138/4/82619/the-chen-primes-contain-arbitrarily-long-arithmetic-progressions">The Chen primes contain arbitrarily long arithmetic progressions</a>, Acta Arithmetica 138 (2009), pp. 301-315.
%e 3, 5, 7 = a(3)
%e 5, 11, 17, 23 = a(4)
%e 5, 11, 17, 23, 29 = a(5)
%e 107, 137, 167, 197, 227, 257 = a(6)
%e 179, 389, 599, 809, 1019, 1229, 1439 = a(7)
%e 881, 1091, 1301, 1511, 1721, 1931, 2141, 2351 = a(8)
%e 4721, 7451, 10181, 12911, 15641, 18371, 21101, 23831, 26561 = a(9)
%e 122069, 124799, 127529, 130259, 132989, 135719, 138449, 141179, 143909, 146639 = a(10)
%e 182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949 = a(11)
%e 182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949, 2062889 = a(12)
%e 205492409, 216843749, 228195089, 239546429, 250897769, 262249109, 273600449, 284951789, 296303129, 307654469, 319005809, 330357149, 341708489 = a(13)
%e 19712507, 176829467, 333946427, 491063387, 648180347, 805297307, 962414267, 1119531227, 1276648187, 1433765147, 1590882107, 1747999067, 1905116027, 2062232987 = a(14)
%o (PARI) primorial(n)=vecprod(primes(primepi(n)));
%o listChen(lim)=my(v=List([2]), semi=List(), L=lim+2, p=3); forprime(q=3, L\3, forprime(r=3, min(L\q, q), listput(semi, q*r))); semi=Set(semi); forprime(q=7, lim, if(setsearch(semi, q+2), listput(v, q))); forprime(q=5, L, if(q-p==2, listput(v, p)); p=q); Set(v)
%o chen=listChen(1e6); \\ Increase as needed to find more terms
%o a(n,startAt=n)=n--; my(div=lcm(primorial(n+1),n)); for(i=startAt,#chen, for(j=1,i-n, my(d=chen[i]-chen[j],g); if(d%div,next); g=d/n; forstep(k=chen[j]+g, chen[i]-g, g, if(!setsearch(chen,k), next(2))); return(chen[i])))
%Y Cf. A109611, A005115.
%K nonn,more
%O 1,1
%A _Charles R Greathouse IV_, Aug 25 2017
%E a(14) from _Charles R Greathouse IV_, Sep 06 2017
|
