%I #13 Dec 07 2019 12:18:26
%S 0,0,1,2,1,2,2,1,3,1,1,2,2,3,2,1,1,4,2,2,3,1,2,4,2,1,3,1,2,3,2,2,4,2,
%T 3,2,0,2,4,3,2,4,1,3,4,1,2,6,2,2,3,2,3,4,1,1,3,3,4,4,2,1,6,1,3,3,1,3,
%U 6,3,1,4,2,4,6,1,3,4,1,4,3,3,4,5,2,3,4
%N Number of partitions of n into a prime and a triangular number.
%C Indices of zeros: 0 followed by A076768.
%H T. D. Noe, <a href="/A224362/b224362.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: (Sum_{i>=0} x^(i*(i+1)/2))*(Sum_{j>=1} x^prime(j)). - _Ilya Gutkovskiy_, Feb 07 2017
%t nn = 13; tri = Table[n*(n + 1)/2, {n, 0, nn}]; pr = Prime[Range[PrimePi[tri[[-1]]]]]; Table[Length[Intersection[pr, n - tri]], {n, 0, tri[[-1]]}] (* _T. D. Noe_, Apr 05 2013 *)
%o (Python)
%o import math
%o primes = [2]*2
%o primes[1] = 3
%o def isprime(k):
%o for p in primes:
%o if k%p==0: return 0
%o primes.append(k)
%o return 1
%o def rootTriangular(a):
%o sr = 2**(int(math.log(a,2))+2)
%o while a < sr*(sr+1)/2:
%o sr>>=1
%o b = sr>>1
%o while b:
%o s = sr+b
%o if a >= s*(s+1)/2:
%o sr = s
%o b>>=1
%o return sr
%o for i in range(1L<<10):
%o k = 0
%o for p in primes:
%o if i <= p: continue
%o r = rootTriangular(i - p)
%o if r*(r+1)/2 == i-p: k+=1
%o if i>1:
%o if i<=3: k += 1
%o else: k += isprime(i)
%o print str(k)+',',
%Y Cf. A000040, A000217, A076768, A101182.
%K nonn
%O 0,4
%A _Alex Ratushnyak_, Apr 04 2013