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%I #17 Dec 25 2022 11:33:18
%S 176,35497,45850,68587,87725,229126,488776,705551,827702,1085876,
%T 1127100,1255380,1732900,1914785,1972840,2453122,2737126,2749297,
%U 2818776,3245026,4598126,5116190,5522882,6180335,6658120,6939126,6958497,7088327,7114437,7140595
%N The first of three consecutive pentagonal numbers the sum of which is equal to the sum of three consecutive primes.
%H Robert Israel, <a href="/A298250/b298250.txt">Table of n, a(n) for n = 1..2352</a>
%e 176 is in the sequence because 176+210+247 (consecutive pentagonal numbers) = 633 = 199+211+223 (consecutive primes).
%p N:= 10^8: # to get all terms where the sums <= N
%p Res:= NULL:
%p mmax:= floor((sqrt(8*N-23)-5)/6):
%p M:= [seq(seq(4*i+j,j=2..3),i=0..mmax/4)]:
%p M3:= map(m -> 9/2*m^2+15/2*m+6, M):
%p for i from 1 to nops(M) do
%p m:= M3[i];
%p r:= ceil((m-8)/3);
%p p1:= prevprime(r+1);
%p p2:= nextprime(p1);
%p p3:= nextprime(p2);
%p while p1+p2+p3 > m do
%p p3:= p2; p2:= p1; p1:= prevprime(p1);
%p od:
%p if p1+p2+p3 = m then
%p Res:= Res, M[i]*(3*M[i]-1)/2;
%p fi
%p od:
%p Res; # Robert Israel, Jan 16 2018
%t Module[{prs3=Total/@Partition[Prime[Range[10^6]],3,1]},Select[ Partition[ PolygonalNumber[ 5,Range[ 5000]],3,1],MemberQ[ prs3,Total[#]]&]][[All,1]] (* _Harvey P. Dale_, Dec 25 2022 *)
%o (PARI) L=List(); forprime(p=2, 8000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(72*t-207, &sq) && (sq-15)%18==0, u=(sq-15)\18; listput(L, (3*u^2-u)/2))); Vec(L)
%Y Cf. A000040, A000326, A054643, A298073, A298168, A298169, A298222, A298223, A298251.
%K nonn
%O 1,1
%A _Colin Barker_, Jan 15 2018
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