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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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 17 05:41 EDT 2021. Contains 345080 sequences. (Running on oeis4.)