A082078 Balanced primes of order three.

17, 53, 157, 173, 193, 229, 349, 439, 607, 659, 701, 709, 977, 1153, 1187, 1301, 1619, 2281, 2287, 2293, 2671, 2819, 2843, 3067, 3313, 3539, 3673, 3727, 3833, 4013, 4051, 4517, 4951, 5101, 5897, 6079, 6203, 6211, 6323, 6679, 6869, 7321, 7589, 7643, 7907

Offset: 1

Author

Labos Elemer, Apr 08 2003

Keywords

• nonn

Comments

The arithmetic mean of 6 primes in its "neighborhood"; not to be confused with 'Triply balanced primes' (A081415).

A balanced prime of order three is not necessarily balanced of order one (A006562) or two (A082077), etc. [Typo corrected by Zak Seidov, Jul 23 2008]

Examples

			p = 53 = (41 + 43 + 47 + 53 + 59 + 61 + 67)/7 = 371/7 i.e. it is the arithmetic mean.
		

Links

• Aaron Toponce, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006562, A082077, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704.

Cf. A096693, A082080, A081415, A051795, A006562.

Programs

GAP

P:=Filtered([1..10000],IsPrime);;
a:=List(Filtered(List([0..1000],k->List([4..10],j->P[j-3+k])), i->
Sum(i)/7=i[4]),m->m[4]); # Muniru A Asiru, Feb 14 2018

Mathematica

Do[s3=Prime[n]+Prime[n+1]+Prime[n+2]; s5=Prime[n-1]+s3+Prime[n+3]; s7=Prime[n-2]+s5+Prime[n+4]; If[Equal[s7/7, Prime[n+1]], Print[Prime[n+1]]], {n, 3, 5000}]
(* Second program: *)
With[{k = 3}, Select[MapIndexed[{Prime[First@ #2 + k], #1} &, Mean /@ Partition[Prime@ Range[10^3], 2 k + 1, 1]], SameQ @@ # &][[All, 1]]]  (* Michael De Vlieger, Feb 15 2018 *)
Select[Partition[Prime[Range[1500]],7,1],Mean[#]==#[[4]]&][[All,4]] (* Harvey P. Dale, Jul 01 2022 *)

PARI

isok(p) = {if (isprime(p), k = primepi(p); if (k > 3, sum(i=k-3, k+3, prime(i)) == 7*p;););} \\ Michel Marcus, Mar 07 2018