A234531 Pentagonal numbers which are the arithmetic mean of two consecutive primes.

12, 176, 376, 532, 590, 3015, 4510, 4676, 7315, 7526, 7957, 8855, 12650, 15555, 17120, 19437, 20126, 22265, 25676, 29330, 30175, 40755, 48510, 54626, 78547, 82017, 91390, 97410, 101270, 102051, 102835, 105205

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

			376 is in the sequence because 376 = 16*(3*16 - 1)/2 = (373 + 379)/2, the arithmetic mean of two consecutive primes.
532 is in the sequence because 532 = 19*(3*19 - 1)/2 = (523 + 541)/2, the arithmetic mean of two consecutive primes.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..5535

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,g;  a:= n*(3*n-1)/2; b:=prevprime(a); d:=nextprime(b); g:=(b+d)/2;  if a=g then RETURN (a);  fi; end: seq(KD(), n = 2..500);

Mathematica

Select[PolygonalNumber[5,Range[300]],!PrimeQ[#]&&#==(NextPrime[ #]+ NextPrime[ #,-1])/2&] (* Harvey P. Dale, Dec 26 2022 *)

PARI

lista(nn) = for (n=1, nn, pn = n*(3*n-1)/2; if (pn > 2, precp = precprime(pn-1); if (pn == (precp + nextprime(precp+1))/2, print1(pn, ", ")))) \\ Michel Marcus, Dec 30 2013

Extensions

Erroneous term 5 removed by Michel Marcus, Dec 30 2013