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

1,1

Comments

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

Links

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

Example

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.

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 *)

Prog

(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

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).

Sequence in context: A099930 A307549 A052208 * A045952 A227714 A358951

Adjacent sequences: A234528 A234529 A234530 * A234532 A234533 A234534

Keyword

nonn

Author

K. D. Bajpai, Dec 27 2013

Extensions

Erroneous term 5 removed by Michel Marcus, Dec 30 2013

Status

approved