%I A117065
%S A117065 19,31,43,67,89,101,113,131,229,241,277,359,383,491,523,619,631,643,701,
%T A117065 761,1321,1381,1621,2221,2861
%N A117065 Primes that are not the sum of 3 pentagonal numbers.
%C A117065 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}.
%C A117065 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 (noe(AT)sspectra.com), Apr 19 2006
%D A117065 R. K. Guy, Every number is expressible as the sum of how many polygonal 
               numbers?, Amer. Math. Monthly 101 (1994), 169-172.
%F A117065 A000040 INTERSECT A003679.
%t A117065 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 (noe(AT)sspectra.com), Apr 19 2006
%Y A117065 Cf. A000040, A000326, A003679, A064826.
%Y A117065 Sequence in context: A040068 A096787 A104006 this_sequence A006035 A104485 
               A141184
%Y A117065 Adjacent sequences: A117062 A117063 A117064 this_sequence A117066 A117067 
               A117068
%K A117065 easy,nonn
%O A117065 1,1
%A A117065 Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 17 2006
%E A117065 More terms from T. D. Noe (noe(AT)sspectra.com), Apr 19 2006