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A185004 Ramanujan modulo primes R_(3,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,1)(x) - pi_(3,1)(x/2) >= n, where pi_(3,1)(x) is the number of primes==1 (mod 3) <= x.

%I A185004 #44 Jun 06 2017 23:28:32
%S A185004 7,31,43,67,97,103,151,163,181,223,229,271,331,337,367,373,409,433,
%T A185004 487,499,571,577,601,607,631,643,709,727,751,769,823,853,883,937,991,
%U A185004 1009,1021,1033,1051,1063,1087,1117,1123,1231,1291,1297,1303
%N A185004 Ramanujan modulo primes R_(3,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,1)(x) - pi_(3,1)(x/2) >= n, where pi_(3,1)(x) is the number of primes==1 (mod 3) <= x.
%C A185004 All terms are primes==1 (mod 3).
%C A185004 A modular generalization of Ramanujan numbers, see Section 6 of the Shevelev-Greathouse-Moses paper.
%C A185004 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 3*k+1.
%C A185004 The function pi_(3,1)(n) starts 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3,... with records occurring as specified in A123365/A002476. - _R. J. Mathar_, Jan 10 2013
%H A185004 G. C. Greubel, <a href="/A185004/b185004.txt">Table of n, a(n) for n = 1..1000</a>
%H A185004 Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%F A185004 lim(a(n)/prime(4*n)) = 1 as n tends to infinity.
%p A185004 pimod := proc(m,n,x)
%p A185004     option remember;
%p A185004     a := 0 ;
%p A185004     for k from n to x by m do
%p A185004         if isprime(k) then
%p A185004             a := a+1 ;
%p A185004         end if;
%p A185004     end do:
%p A185004     a ;
%p A185004 end proc:
%p A185004 a := [seq(0,n=1..100)] ;
%p A185004 for x from 1 do
%p A185004     pdiff := pimod(3,1,x)-pimod(3,1,x/2) ;
%p A185004     if pdiff+1 <= nops(a) then
%p A185004         v := x+1 ;
%p A185004         n := pdiff+1 ;
%p A185004         if n<v then
%p A185004             a := subsop(n=v,a) ;
%p A185004             print(a) ;
%p A185004         end if;
%p A185004     end if;
%p A185004 end do: # _R. J. Mathar_, Jan 10 2013
%t A185004 max = 100; pimod[m_, n_, x_] := pimod[m, n, x] = Module[{a = 0}, For[k = n, k <= x, k = k + m, If[PrimeQ[k], a = a + 1]]; a]; a[_] = 0; For[x = 1, x <= max^2, x++, pdiff = pimod[3, 1, x] - pimod[3, 1, x/2]; If[ pdiff + 1 <= max, v = x + 1; n = pdiff + 1; If[ n < v , a[n] = v ] ] ]; Table[a[n], {n, 1, max}] (* _Jean-François Alcover_, Jan 28 2013, translated and adapted from _R. J. Mathar_'s Maple program *)
%Y A185004 Cf. A104272, A185005, A185006, A185007.
%K A185004 nonn
%O A185004 1,1
%A A185004 _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 18 2012