A219478 -id:A219478 - cp's OEIS frontend

A082077 Balanced primes of order two.

79, 281, 349, 439, 643, 677, 787, 1171, 1733, 1811, 2141, 2347, 2389, 2767, 2791, 3323, 3329, 3529, 3929, 4157, 4349, 4751, 4799, 4919, 4951, 5003, 5189, 5323, 5347, 5521, 5857, 5861, 6287, 6337, 6473, 6967, 6997, 7507, 7933, 8233, 8377, 8429, 9377, 9623, 9629, 10093, 10333

Offset: 1

Labos Elemer, Apr 08 2003

nonn

The arithmetic mean of 4 primes in its "neighborhood"; not to be confused with 'Doubly balanced primes' (A051795).
Balanced primes of order two are not necessarily balanced of order one (A006562) or three (A082078).
Subsequence of A219478, Peter Schorn, May 01 2025

			p = 79 = (71 + 73 + 79 + 83 + 89)/5 = 395/5 i.e. it is both the arithmetic mean and median.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006562, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096693, A082080, A081415, A051795, A006562, A219478.

Do[s3=Prime[n]+Prime[n+1]+Prime[n+2]; s5=Prime[n-1]+s3+Prime[n+3]; If[Equal[s5/5, Prime[n+1]], Print[Prime[n+1]]], {n, 3, 3000}]
Select[Partition[Prime[Range[1500]],5,1],Mean[#]==#[[3]]&][[All,3]] (* Harvey P. Dale, Nov 04 2019 *)

p=2;q=3;r=5;s=7;forprime(t=11,1e9,if(p+q+s+t==4*r,print1(r", ")); p=q; q=r; r=s; s=t) \\ Charles R Greathouse IV, Nov 20 2012

A219505 Numbers that are arithmetic mean of 5 successive primes.

Original entry on oeis.org

79, 281, 295, 329, 349, 355, 403, 439, 511, 643, 677, 685, 787, 805, 841, 887, 949, 963, 1171, 1179, 1195, 1241, 1327, 1339, 1397, 1435, 1463, 1503, 1547, 1641, 1685, 1717, 1733, 1779, 1811, 1917, 1957, 1991, 2033, 2061, 2141, 2147, 2347, 2389, 2395, 2433

Offset: 1

Zak Seidov, Nov 21 2012

nonn

			a(1) = A219478(1) = 79 = (prime(20)+...+prime(24))/5 = (71+ 73+ 79+ 83+ 89)/5.

Zak Seidov, Table of n, a(n) for n = 1..1000

A219478 is subsequence.

Select[(Mean /@ Partition[Prime[Range[1000]], 5, 1]), IntegerQ]

A357133 a(n) is the least prime that is the arithmetic mean of n consecutive primes.

Original entry on oeis.org

5, 127, 79, 101, 17, 269, 491, 727, 53, 23, 71, 181, 29, 31, 37, 43, 563, 331, 883, 283, 173, 307, 157, 113, 353, 571, 347, 89, 263, 139, 179, 1201, 281, 1553, 137, 5167, 347, 563, 2083, 2087, 491, 1867, 353, 463, 1973, 199, 599, 4373, 149, 9929, 277, 463, 1259, 251, 397, 2897, 787, 263, 2161

Offset: 3

J. M. Bergot and Robert Israel, Sep 14 2022

nonn

			a(5) = 79 because 79 is the average of the 5 consecutive primes 71, 73, 79, 83, 89, and 79 is the least prime that works.

Robert Israel, Table of n, a(n) for n = 3..7000

Cf. A006562, A126096, A219478, A122531

P:= [seq(ithprime(i),i=1..10^5)]:
S:= ListTools:-PartialSums(P):
g:= proc(n) local k,r;
   for k from 1 to 10^5-n do
      r:= (S[k+n]-S[k])/n;
      if r::integer and isprime(r) then return r fi
   od;
   -1
end proc:
map(g, [$3..100]);

a={}; n=3; For[k=1, k<=1200, k++, If[PrimeQ[p=Sum[Prime[k+i], {i,0,n-1}]/n], AppendTo[a,p]; n++; k=1]]; a (* Stefano Spezia, Sep 15 2022 *)

from itertools import islice
from sympy import isprime, nextprime
def agen():
    n, plst0 = 3, [2, 3, 5]
    while True:
        plst = plst0[:]
        while True:
            q, r = divmod(sum(plst), n)
            if r == 0 and isprime(q): yield q; break
            plst = plst[1:] + [nextprime(plst[-1])]
        plst0.append(nextprime(plst0[-1]))
        n += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Sep 14 2022

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A082077 Balanced primes of order two.

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A219505 Numbers that are arithmetic mean of 5 successive primes.

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A357133 a(n) is the least prime that is the arithmetic mean of n consecutive primes.

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