login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A124661 Popular primes: primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for all k = 1,2,...n-2. 7
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 * A134266 A233043 A231099

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 12 01:08 EST 2017. Contains 295936 sequences.