A381458 Primes p such that p/prev_prime(p) < 1 + (1/PrimePi(p)).

19, 31, 43, 61, 73, 103, 109, 139, 151, 167, 181, 193, 197, 199, 227, 229, 233, 241, 271, 281, 283, 311, 313, 317, 349, 353, 383, 401, 421, 433, 443, 461, 463, 467, 491, 503, 523, 571, 601, 617, 619, 643, 647, 661, 677, 743, 761, 773, 811, 823, 827, 829, 857, 859, 863, 881, 883, 887, 911, 941, 971

Offset

1,1

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 3/5.

Example

19 = prime(8) following 17 = prime(7) is a term because 19/17 < 1 + 1/8.

Maple

q:= p-> isprime(p) and p/prevprime(p) < 1+1/numtheory[pi](p):
select(q, [$3..1000])[];  # Alois P. Heinz, Mar 21 2025

Mathematica

Select[Prime[Range[2, 165]], (#/NextPrime[#, -1])<1+(1/PrimePi[#])&] (* James C. McMahon, Mar 29 2025 *)

Prog

(PARI) my(N=3); forprime(P=5, 1000, my(Q=precprime(P-1), AR0=1+(1/N), AR=P/Q); N++; if(AR<AR0, print1(P, ", ")));

Crossrefs

Cf. A000720, A151799.

Sequence in context: A043298 A286313 A040068 * A096787 A104006 A117065

Adjacent sequences: A381455 A381456 A381457 * A381459 A381460 A381461

Keyword

nonn

Author

Alain Rocchelli, Feb 28 2025

Status

approved