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.

7, 31, 43, 67, 97, 103, 151, 163, 181, 223, 229, 271, 331, 337, 367, 373, 409, 433, 487, 499, 571, 577, 601, 607, 631, 643, 709, 727, 751, 769, 823, 853, 883, 937, 991, 1009, 1021, 1033, 1051, 1063, 1087, 1117, 1123, 1231, 1291, 1297, 1303

Offset: 1

Author

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

Keywords

• nonn

Comments

All terms are primes==1 (mod 3).

A modular generalization of Ramanujan numbers, see Section 6 of the Shevelev-Greathouse-Moses paper.

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.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• 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, A185005, A185006, A185007.

Programs

Maple

pimod := proc(m,n,x)
    option remember;
    a := 0 ;
    for k from n to x by m do
        if isprime(k) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
a := [seq(0,n=1..100)] ;
for x from 1 do
    pdiff := pimod(3,1,x)-pimod(3,1,x/2) ;
    if pdiff+1 <= nops(a) then
        v := x+1 ;
        n := pdiff+1 ;
        if nR. J. Mathar, Jan 10 2013

Mathematica

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 *)

Formula

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