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A124661 Popular primes: primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for all k = 1,2,...n-2. 8
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 (list; graph; refs; listen; history; text; internal format)
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: A118371 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

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Last modified December 14 05:17 EST 2018. Contains 318090 sequences. (Running on oeis4.)