A028388 Good primes (version 2): prime(n) such that prime(n)^2 > prime(n-i)*prime(n+i) for all 1 <= i <= n-1.

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967, 1009, 1031, 1087, 1151, 1213, 1277

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

Selfridge conjectured, and Pomerance proved, that there are infinitely many numbers in this sequence. Pomerance asks if the sequence has density 0. - Charles R Greathouse IV, Apr 14 2011

References

• Guy, R. K. `Good' Primes and the Prime Number Graph. A14 in Unsolved Problems in Number Theory, 2nd ed. Springer-Verlag, pp. 32-33, 1994.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Carl Pomerance, The prime number graph, Mathematics of Computation 33:145 (1979), pp. 399-408.
• Eric Weisstein's World of Mathematics, Good Prime
• Eric Weisstein's World of Mathematics, Selfridge's Conjecture

Crossrefs

Cf. A046869.

Programs

Magma

[NthPrime(n): n in [2..220] | forall{i: i in [1..n-1] | NthPrime(n)^2 gt NthPrime(n-i)*NthPrime(n+i)}]; // Bruno Berselli, Oct 23 2012

Mathematica

Module[{nn=300,prs},prs=Prime[Range[2nn]];qprQ[n_]:=Module[{pi= PrimePi[n]}, n^2>Max[Times@@@Thread[{Take[prs,pi-1],Reverse[Take[ prs,{pi+1,2 pi-1}]]}]]];Select[Take[prs,nn],qprQ]] (* Harvey P. Dale, May 13 2012 *)

PARI

is(n)=if(!isprime(n),return(0));my(p=n,q=n,n2=n^2);while(p>2, p=precprime(p-1); q=nextprime(q+1); if(n22 \\ Charles R Greathouse IV, Jul 02 2013