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A298463 The first of two consecutive pentagonal numbers the sum of which is equal to the sum of two consecutive primes.

%I A298463 #19 Jan 20 2024 16:14:20
%S A298463 70,3577,10795,36895,55777,70525,78547,125137,178365,208507,258130,
%T A298463 329707,349692,394497,438751,468442,478555,499105,619852,663005,
%U A298463 753667,827702,877455,900550,1025480,1085876,1169092,1201090,1211852,1233520,1339065,1508512
%N A298463 The first of two consecutive pentagonal numbers the sum of which is equal to the sum of two consecutive primes.
%H A298463 Chai Wah Wu, <a href="/A298463/b298463.txt">Table of n, a(n) for n = 1..10000</a>
%e A298463 70 is in the sequence because 70+92 (consecutive pentagonal numbers) = 162 = 79+83 (consecutive primes).
%t A298463 Select[Partition[PolygonalNumber[5,Range[1500]],2,1],CompositeQ[Total[#]/2]&&Total[#] == NextPrime[ Total[#]/2]+NextPrime[Total[#]/2,-1]&][[;;,1]] (* _Harvey P. Dale_, Jan 20 2024 *)
%o A298463 (PARI) L=List(); forprime(p=2, 1600000, q=nextprime(p+1); t=p+q; if(issquare(12*t-8, &sq) && (sq-2)%6==0, u=(sq-2)\6; listput(L, (3*u^2-u)/2))); Vec(L)
%o A298463 (Python)
%o A298463 from __future__ import division
%o A298463 from sympy import prevprime, nextprime
%o A298463 A298463_list, n, m = [], 1 ,6
%o A298463 while len(A298463_list) < 10000:
%o A298463     k = prevprime(m//2)
%o A298463     if k + nextprime(k) == m:
%o A298463         A298463_list.append(n*(3*n-1)//2)
%o A298463     n += 1
%o A298463     m += 6*n-1 # _Chai Wah Wu_, Jan 20 2018
%Y A298463 Cf. A000040, A000326, A001043, A061275, A298462, A298464, A298465, A298466.
%K A298463 nonn
%O A298463 1,1
%A A298463 _Colin Barker_, Jan 19 2018