A124661 Primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for k = 1..n-2.

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 73, 83, 109, 113, 181, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789

Offset: 1

Author

Artur Jasinski, Dec 23 2006

Keywords

• nonn
• easy

Comments

The first two primes, 2 and 3, are tested against an empty set of k, and we include them, defining such a test to have a positive outcome.

McNew's "popular primes" sequence (A385503) has the same first 14 terms, differing first by excluding 181. McNew says that a prime p is "popular" on an interval [2, k] if no prime occurs more frequently than p as the greatest prime factor (gpf, A006530) of the integers in that interval. - N. J. A. Sloane, Jul 25 2017 and Peter Munn, Jul 01 2025

See the Pomerance link for a proof that the sequence is infinite. - Peter Munn, Jul 01 2025

Examples

			prime(11)=31 is in the sequence because prime(10)+prime(12) = 66, prime(9)+prime(13) = 64,..., prime(2)+prime(20) = 74 are all >= 62 = 2*31.
prime(10) = 29 is not in the sequence because prime(9)+prime(11) = 54 for example is smaller than 58 = 2*29.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Nathan McNew, Popular values of the largest prime divisor function, arXiv:1504.05985 [math.NT], 2015.
• Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
• C. Pomerance, The prime number graph, Math. Comp. 33 (1979) 399--408. - _Nathan McNew_, Apr 04 2014

Crossrefs

Cf. A006530, A051635, A385503.

Programs

Mathematica

Select[Prime@ Range@ 300, Function[{p, n}, NoneTrue[Range[n - 2], Prime[n - #] + Prime[n + #] < 2 p &]] @@ {#, PrimePi@ #} &] (* Michael De Vlieger, Jul 25 2017 *)

PARI

isok(p) = {n = primepi(p); for (k=1, n-2, if (prime(n-k) + prime(n+k) < 2*p, return (0));); return (1);}
lista(nn) = {for(n=1, nn, if (isok(prime(n)), print1(prime(n), ", ");););} \\ Michel Marcus, Nov 03 2013

Python

from sympy import prime
A124661_list = []
for n in range(1,10**6):
    p = prime(n)
    for k in range(1,n-1):
        if prime(n-k)+prime(n+k) < 2*p:
            break
    else:
        A124661_list.append(p) # Chai Wah Wu, Jul 25 2017

Extensions

Sequence extended by R. J. Mathar, Mar 28 2010

Edited, restoring previous name, by Peter Munn, Jul 01 2025