OFFSET
1,1
COMMENTS
All terms are primes==1 (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+1.
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
FORMULA
lim(a(n)/prime(4*n)) = 1 as n tends to infinity.
MATHEMATICA
Table[1 + NestWhile[#1 - 1 &, A104272[[3 k]], Count[Mod[Select[Range@@{Floor[#1/2 + 1], #1}, PrimeQ], 4], 1] >= 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[s < nn, A104272[[s + 1]] = k], {k, Prime[3*nn]}]; A104272 = A104272 + 1 (* T. D. Noe, Nov 15 2010 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012
STATUS
approved