A124661 Popular primes: primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for all k = 1,2,...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

1,1

Comments

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

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.

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

Example

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.

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 *)

Prog

(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)
def a124661(end):
....a124661_list=[2, 3, 5, 7]
....primes=[2, 3]
....add=True
....for num in range(4, end*3):
........prime=False
........length=len(primes)
........for y in range(0, length):
............if num % primes[y]!=0:
................prime=True
............else:
................prime=False
................break
........if (prime):
............primes.append(num)
....for x in range(4, int(len(primes)/2)):
........for k in range(1, x-2):
............if (primes[x-k]+primes[x+k]>=primes[x]*2):
................add=True
............else:
................add=False
................break
........if (add):
............if (primes[x]>end):
................break
............else:
................a124661_list.append(primes[x])
....return a124661_list
# Conner L. Delahanty, Apr 19 2014
(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

Crossrefs

Cf. A006530, A051635.

Sequence in context: A330968 A153591 A038917 * A319126 A134266 A233043

Adjacent sequences: A124658 A124659 A124660 * A124662 A124663 A124664

Keyword

nonn,easy

Author

Artur Jasinski, Dec 23 2006

Extensions

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

Status

approved