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.

852, 172212, 272001, 341055, 777240, 1451892, 1710402, 2361910, 2922526, 3950382, 4794522, 5414050, 6272015, 7989142, 10580176, 10780301, 11325882, 12173777, 12483395, 14432055, 14665630, 15890910, 16581775, 16962972, 17124772

Offset: 1

Author

K. D. Bajpai, Dec 27 2013

Keywords

• nonn
• less

Comments

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

Examples

			852 is in the sequence because 852 = 24*(3*24-1)/2 = (839 + 853 + 857 + 859)/4, the arithmetic mean of four consecutive primes.
172212 is in the sequence because 172212 = 339*(3*339-1)/2 = (172199 +172213+ 172217 + 172219)/4, the arithmetic mean of four consecutive primes.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..1786

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,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);

Mathematica

Mean/@Select[Partition[Prime[Range[2*10^6]],4,1],With[{m=Mean[#]},IntegerQ[(1+Sqrt[1+24m])/6]&&#[[1]]Harvey P. Dale, May 12 2025 *)

Extensions

Typo in definition corrected by N. J. A. Sloane, May 13 2025 at the suggestion of Harvey P. Dale.