A164294 Primes prime(k) such that all integers in [(prime(k-1)+1)/2,(prime(k)-1)/2] are composite, excluding those primes in A080359.

131, 151, 229, 233, 311, 571, 643, 727, 941, 1013, 1051, 1153, 1373, 1531, 1667, 1669, 1723, 1783, 1787, 1831, 1951, 1979, 2029, 2131, 2213, 2239, 2311, 2441, 2593, 2621, 2633, 2659, 2663, 2887, 3001, 3011, 3019, 3121, 3169, 3209, 3253, 3347, 3413, 3457

Offset

1,1

Comments

The primes of A080359 larger than 3 all have the property that the integers in the interval selected by halving the value of the preceding prime and halving their own value are all composite. This sequence here collects the primes that are not in A080359 but still share this property of the prime-free subinterval.

Links

Jean-François Alcover, Table of n, a(n) for n = 1..1106

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009. [From Vladimir Shevelev, Aug 20 2009]

V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.

Formula

A164333 \ A080359.

Example

For the prime 1531=A000040(242), the preceding prime is A000040(241)=1523, and the integers from (1523+1)/2 = 762 up to (1531-1)/2 = 765 are all composite, as they fall in the gap between A000040(135) and A000040(136). In addition, 1531 is not in A080359, which adds 1531 to this sequence here.

Mathematica

maxPrime = 3500;
kmax = PrimePi[maxPrime];
A164333 = Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ] &][[All, 2]]*2 + 1;
b[1] = 2; b[n_] := b[n] = Module[{k = b[n - 1]}, While[(PrimePi[k] - PrimePi[Quotient[k, 2]]) != n, k++]; k];
A080359 = Reap[For[n = 1, b[n] <= maxPrime, n++, Sow[b[n]]]][[2, 1]];
Complement[A164333, A080359] (* Jean-François Alcover, Sep 14 2018 *)

Prog

(PARI) okprime(p) = { my(k = primepi(p)); for (i = (prime(k-1)+1)/2, (prime(k)-1)/2, if (isprime(i), return (0)); ); return (1); }
lista(nn) = {vlp = readvec("b080359.txt"); forprime (p=2, nn, if (! vecsearch(vlp, p) && okprime(p), print1(p, ", ")); ); } \\ Michel Marcus, Jan 15 2014

Crossrefs

Cf. A080359, A104272, A164288, A001262, A001567, A062568, A141232, A164368.

Sequence in context: A273049 A134951 A173071 * A155924 A270237 A090264

Adjacent sequences: A164291 A164292 A164293 * A164295 A164296 A164297

Keyword

nonn

Author

Vladimir Shevelev, Aug 12 2009

Extensions

Extended beyond 571 by R. J. Mathar, Oct 02 2009

Status

approved