A234297 -id:A234297 - cp's OEIS frontend

A242218 Semiprimes which are the arithmetic mean of three consecutive primes.

511, 537, 1073, 1461, 1501, 1541, 1763, 2071, 2181, 2449, 4101, 4387, 4399, 4467, 4559, 4607, 4681, 4705, 5089, 5257, 5429, 6415, 6621, 6671, 7097, 7111, 7261, 7391, 7447, 7811, 7831, 7897, 7909, 7969, 8079, 8129, 8193, 8333, 8639, 8915, 9101, 9113, 9123, 9211

Offset: 1

K. D. Bajpai, May 07 2014

nonn

			a(1) = 511 = (503 + 509 + 521)/3 = 7 * 73 is semiprime.
a(2) = 537 = (523 + 541 + 547)/3 = 3 * 179 is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A133529, A216432, A234297.

with(numtheory): A242218:= proc()local k ; k:=(ithprime(x)+ithprime(x+1)+ithprime(x+2))/3; if k=floor(k) and bigomega(k)=2 then RETURN (k);  fi; end: seq(A242218 (),x=2..2000);

for(k=1, 2000, t=prime(k)+prime(k+1)+prime(k+2); if(t%3==0 && bigomega(t/3)==2, print1(t/3, ", "))) \\ Colin Barker, May 08 2014

A234318 Squares t^2 = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < t^2 < q < r < s.

Original entry on oeis.org

15876, 35721, 59049, 65025, 488601, 828100, 1144900, 3857296, 4822416, 4901796, 5107600, 5322249, 5856400, 6100900, 6760000, 10536516, 11716929, 12503296, 13468900, 14197824, 14638276, 15163236, 18748900, 21455424, 22127616, 22638564, 24049216, 24098281, 24108100

Offset: 1

K. D. Bajpai, Dec 23 2013

nonn

			15876 is in the sequence because 15876 = 126^2 = (15859+15877+15881+15887)/4, the arithmetic mean of four consecutive primes.
35721 is in the sequence because 35721 = 189^2 = (35677+35729+35731+35747)/4, the arithmetic mean of four consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..2869

Cf. A000290 (squares: a(n) = n^2).
Cf. A062703 (squares: sum of two consecutive primes).
Cf. A069495 (squares: arithmetic mean of two consecutive primes).
Cf. A234240 (cubes: arithmetic mean of three consecutive primes).
Cf. A234297 (squares: arithmetic mean of three consecutive primes).

KD := proc() local a,b,d,e,f,g; a:=n^2;b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=nextprime(e); g:=(b+d+e+f)/4; if a=g then RETURN (a); fi; end: seq(KD(), n=2..10000);

fcpQ[{a_,b_,c_,d_}]:=Module[{m=Mean[{a,b,c,d}]},IntegerQ[ Sqrt[ m]] && a< m< b]; Mean/@Select[Partition[Prime[Range[1600000]],4,1],fcpQ] (* Harvey P. Dale, Apr 24 2017 *)

list(lim)=my(v=List(), p=2, q=3, r=5, t); forprime(s=7, nextprime(nextprime(lim+1)+1), t=(p+q+r+s)/4; if(denominator(t)==1 && issquare(t) && t < q, listput(v, t)); p=q; q=r; r=s); Vec(v) \\ Charles R Greathouse IV, Jan 03 2014

Definition corrected by Michel Marcus and Charles R Greathouse IV, Jan 03 2014

cp's OEIS Frontend

A242218 Semiprimes which are the arithmetic mean of three consecutive primes.

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