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A298250 The first of three consecutive pentagonal numbers the sum of which is equal to the sum of three consecutive primes.

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