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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.

%I A234532 #12 Jun 25 2022 17:08:10
%S A234532 9087,29751,291501,602617,1505505,1778337,1941997,2137857,3032415,
%T A234532 4629695,5016947,5038917,7837551,8030737,9328807,11935651,19158427,
%U A234532 35616757,40964001,41073817,42594697,44289817,56141827,59267551
%N 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.
%C A234532 The n-th pentagonal number is (3*n^2 - n)/2 = n*(3*n - 1)/2.
%H A234532 K. D. Bajpai, <a href="/A234532/b234532.txt">Table of n, a(n) for n = 1..3747</a>
%e A234532 9087 is in the sequence because 9087 = 78 *(3*78 - 1)/2 = (9067 + 9091 + 9103)/3, the arithmetic mean of three consecutive primes.
%e A234532 29751 is in the sequence because 29751 = 141*(3*141 - 1)/2 = (29741 + 29753 + 29759)/3, the arithmetic mean of three consecutive primes.
%p A234532 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);
%Y A234532 Cf. A000326 (pentagonal numbers:  n * (3*n - 1)/2).
%Y A234532 Cf. A069495 (squares: arithmetic mean of two consecutive primes).
%Y A234532 Cf. A234240 (cubes: arithmetic mean of three consecutive primes).
%K A234532 nonn
%O A234532 1,1
%A A234532 _K. D. Bajpai_, Dec 27 2013