OFFSET
1,1
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
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)
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
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Dec 23 2006
EXTENSIONS
Sequence extended by R. J. Mathar, Mar 28 2010
Edited, restoring previous name, by Peter Munn, Jul 01 2025
STATUS
approved