A117065 Primes that are not the sum of 3 pentagonal numbers.

19, 31, 43, 67, 89, 101, 113, 131, 229, 241, 277, 359, 383, 491, 523, 619, 631, 643, 701, 761, 1321, 1381, 1621, 2221, 2861

Offset: 1

Author

Jonathan Vos Post, Apr 17 2006

Keywords

• more
• hard
• nonn

Comments

5 is the only prime pentagonal number; every greater pentagonal number A000326(n) = n(3n-1)/2 is either divisible by n/2 or (3n-1)/2. Every number is the sum of 5 pentagonal numbers, hence every prime is the sum of 5 pentagonal numbers. There are an infinite number of primes which are the sum of two pentagonal numbers, the subset of primes which are the sum of two pentagonal numbers in exactly two different ways begins {211, 853, 1259, 1427, 1571, 2297, 2351}.

The sum may include the pentagonal number 0. Hence this sequence does not have any primes that are the sum of two positive pentagonal numbers. The sequence is probably finite. There are no other primes < 59900. - T. D. Noe, Apr 19 2006

The next term, if it exists, is greater than 160000000. - Jack W Grahl, Jul 10 2018

a(26) > 10^11, if it exists. - Giovanni Resta, Jul 13 2018

Links

• J. W. Grahl, C code which was used to check for elements of this sequence up to 160,000,000.
• R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Crossrefs

Cf. A000040, A000326, A003679, A064826.

Programs

Mathematica

nn=201; pen=Table[n(3n-1)/2, {n,0,nn-1}]; ps=Prime[Range[PrimePi[pen[[ -1]]]]]; Do[n=pen[[i]]+pen[[j]]+pen[[k]]; If[n<=pen[[ -1]]&&PrimeQ[n], ps=DeleteCases[ps, ?(#==n&)]], {i,nn}, {j,i,nn}, {k,j,nn}]; ps (* _T. D. Noe, Apr 19 2006 *)

Formula

A000040 INTERSECT A003679.

Extensions

More terms from T. D. Noe, Apr 19 2006

Mathematica program corrected by Robert Price, Aug 25 2019