A234532 Pentagonal numbers penta(n) = (p + q + r)/3 which are the arithmetic mean of three consecutive primes such that p < penta(n) < q < r.

9087, 29751, 291501, 602617, 1505505, 1778337, 1941997, 2137857, 3032415, 4629695, 5016947, 5038917, 7837551, 8030737, 9328807, 11935651, 19158427, 35616757, 40964001, 41073817, 42594697, 44289817, 56141827, 59267551

Offset: 1

Author

K. D. Bajpai, Dec 27 2013

Keywords

• nonn

Comments

The n-th pentagonal number is (3*n^2 - n)/2 = n*(3*n - 1)/2.

Examples

			9087 is in the sequence because 9087 = 78 *(3*78 - 1)/2 = (9067 + 9091 + 9103)/3, the arithmetic mean of three consecutive primes.
29751 is in the sequence because 29751 = 141*(3*141 - 1)/2 = (29741 + 29753 + 29759)/3, the arithmetic mean of three consecutive primes.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..3747

Crossrefs

Cf. A000326 (pentagonal numbers: n * (3*n - 1)/2).

Cf. A069495 (squares: arithmetic mean of two consecutive primes).

Cf. A234240 (cubes: arithmetic mean of three consecutive primes).

Programs

Maple

KD := proc() local a,b,d,e,g; a:= n*(3*n-1)/2; b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); g:=(b+d+e)/3;  if a=g then RETURN (a); fi; end: seq(KD(), n=2..10000);