A185007 Ramanujan primes R_(4,3)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,3)(x) - pi_(4,3)(x/2) >= n, where pi_(4,3)(x) is the number of primes==3 (mod 4) <= x.

7, 23, 47, 67, 71, 103, 127, 167, 179, 191, 223, 227, 263, 307, 359, 367, 431, 463, 479, 487, 491, 547, 571, 587, 599, 631, 643, 647, 719, 739, 787, 811, 823, 839, 887, 907, 1019, 1031, 1051, 1063, 1087, 1151, 1223, 1279, 1303, 1319, 1399

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012

Keywords

• nonn

Comments

All terms are primes==3 (mod 4).

A general conception of generalized Ramanujan numbers, see in Section 6 of the Shevelev, Greathouse IV, & Moses link.

We conjecture that for all n >= 1, a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 4*k+3.

Links

• Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Crossrefs

Cf. A104272, A185004, A185005, A185006.

Programs

Mathematica

Table[1+NestWhile[#1-1&,A104272[[3 k]],Count[Mod[Select[Range@@{Floor[#1/2+1],#1},PrimeQ],4],3]>=k&],{k,1,10}] using the code nn=1000;A104272=Table[0,{nn}];s=0;Do[If[PrimeQ[k],s++];If[PrimeQ[k/2],s--];If[sA104272[[s+1]]=k],{k,Prime[3*nn]}];A104272=A104272+1 (* T. D. Noe, Nov 15 2010 *)

Formula

lim(a(n)/prime(4*n)) = 1 as n tends to infinity.