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

1,1

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

Formula

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

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 n<v then
            a := subsop(n=v, a) ;
            print(a) ;
        end if;
    end if;
end do: # R. 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 *)

Crossrefs

Cf. A104272, A185005, A185006, A185007.

Sequence in context: A171733 A128028 A000921 * A172490 A349666 A298039

Adjacent sequences: A185001 A185002 A185003 * A185005 A185006 A185007

Keyword

nonn

Author

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

Status

approved