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A291525 a(n) is the largest number in an n-term AP of Chen primes. 1
2, 3, 7, 23, 29, 257, 1439, 2351, 26561, 146639, 1891949, 2062889, 341708489, 2062232987 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..14.

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux 18:1 (2006), pp. 147-182. arXiv:math/0405581 [math.NT]

Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.

Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138 (2009), pp. 301-315.

EXAMPLE

3, 5, 7 = a(3)

5, 11, 17, 23 = a(4)

5, 11, 17, 23, 29 = a(5)

107, 137, 167, 197, 227, 257 = a(6)

179, 389, 599, 809, 1019, 1229, 1439 = a(7)

881, 1091, 1301, 1511, 1721, 1931, 2141, 2351 = a(8)

4721, 7451, 10181, 12911, 15641, 18371, 21101, 23831, 26561 = a(9)

122069, 124799, 127529, 130259, 132989, 135719, 138449, 141179, 143909, 146639 = a(10)

182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949 = a(11)

182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949, 2062889 = a(12)

205492409, 216843749, 228195089, 239546429, 250897769, 262249109, 273600449, 284951789, 296303129, 307654469, 319005809, 330357149, 341708489 = a(13)

19712507, 176829467, 333946427, 491063387, 648180347, 805297307, 962414267, 1119531227, 1276648187, 1433765147, 1590882107, 1747999067, 1905116027, 2062232987 = a(14)

PROG

(PARI) primorial(n)=vecprod(primes(primepi(n)));

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)

chen=listChen(1e6); \\ Increase as needed to find more terms

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])))

CROSSREFS

Cf. A109611, A005115.

Sequence in context: A113872 A120302 A093363 * A248346 A236514 A211997

Adjacent sequences: A291522 A291523 A291524 * A291526 A291527 A291528

KEYWORD

nonn,more

AUTHOR

Charles R Greathouse IV, Aug 25 2017

EXTENSIONS

a(14) from Charles R Greathouse IV, Sep 06 2017

STATUS

approved

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Last modified December 3 17:26 EST 2022. Contains 358535 sequences. (Running on oeis4.)