A363188 Prime numbers that are the exact average of four consecutive odd semiprimes.

53, 67, 89, 199, 223, 277, 349, 439, 449, 461, 487, 491, 499, 523, 557, 569, 643, 683, 877, 883, 929, 941, 1069, 1153, 1259, 1361, 1471, 1487, 1733, 1787, 1901, 1933, 1951, 2111, 2129, 2251, 2297, 2311, 2371, 2521, 2557, 2689, 2777, 2797, 2861, 2917, 2939, 3037, 3041, 3253, 3259, 3271, 3407

Offset: 1

Author

Elmo R. Oliveira, May 20 2023

Keywords

• nonn

Examples

			53 is a term because (49 + 51 + 55 + 57)/4 = 53 is prime.
67 is a term because (57 + 65 + 69 + 77)/4 = 67 is prime.
		

Crossrefs

Cf. A000040, A046315.

Cf. A363074, A363187.

Programs

Maple

OP:= select(isprime,[seq(i,i=3..10000,2)]):
OSP:= sort(select(`<=`,[seq(seq(OP[i]*OP[j],j=1..i),i=1..nops(OP))],3*OP[-1])):
SA:= [seq(add(OSP[i+j],j=0..3)/4,i=1..nops(OSP)-3)]:
select(t -> t::integer and isprime(t), SA); # Robert Israel, May 21 2023

Mathematica

Select[Plus @@@ Partition[Select[Range[1, 3500, 2], PrimeOmega[#] == 2 &], 4, 1] / 4, PrimeQ] (* Amiram Eldar, May 21 2023 *)

Python

from itertools import count, islice
from sympy import factorint, isprime
def semiprime(n): return sum(e for e in factorint(n).values()) == 2
def nextoddsemiprime(n): return next(k for k in count(n+1+(n&1), 2) if semiprime(k))
def agen(): # generator of terms
    osp = [9, 15, 21, 25]
    while True:
        q, r = divmod(sum(osp), len(osp))
        if r == 0 and isprime(q):
            yield q
        osp = osp[1:] + [nextoddsemiprime(osp[-1])]
print(list(islice(agen(), 53))) # Michael S. Branicky, May 21 2023