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A234539 Pentagonal numbers P = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < P < q < r < s.

%I A234539 #25 May 13 2025 12:00:17
%S A234539 852,172212,272001,341055,777240,1451892,1710402,2361910,2922526,
%T A234539 3950382,4794522,5414050,6272015,7989142,10580176,10780301,11325882,
%U A234539 12173777,12483395,14432055,14665630,15890910,16581775,16962972,17124772
%N A234539 Pentagonal numbers P = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < P < q < r < s.
%C A234539 The n-th pentagonal number is (3*n^2 - n)/2 = n*(3*n - 1)/2.
%H A234539 K. D. Bajpai, <a href="/A234539/b234539.txt">Table of n, a(n) for n = 1..1786</a>
%e A234539 852 is in the sequence because 852 = 24*(3*24-1)/2 = (839 + 853 + 857 + 859)/4, the arithmetic mean of four consecutive primes.
%e A234539 172212 is in the sequence because 172212 = 339*(3*339-1)/2 = (172199 +172213+ 172217 + 172219)/4, the arithmetic mean of four consecutive primes.
%p A234539 KD := proc() local a,b,d,e,f,g; a:= n*(3*n-1)/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..5000);
%t A234539 Mean/@Select[Partition[Prime[Range[2*10^6]],4,1],With[{m=Mean[#]},IntegerQ[(1+Sqrt[1+24m])/6]&&#[[1]]<m<#[[2]]&]] (* _Harvey P. Dale_, May 12 2025 *)
%Y A234539 Cf. A000326 (pentagonal numbers: n*(3*n-1)/2).
%Y A234539 Cf. A069495 (squares: arithmetic mean of two consecutive primes).
%Y A234539 Cf. A234240 (cubes: arithmetic mean of three consecutive primes).
%K A234539 nonn,less
%O A234539 1,1
%A A234539 _K. D. Bajpai_, Dec 27 2013
%E A234539 Typo in definition corrected by _N. J. A. Sloane_, May 13 2025 at the suggestion of _Harvey P. Dale_.