cp's OEIS Frontend

A224362 Number of partitions of n into a prime and a triangular number.

%I A224362 #17 Nov 14 2024 10:58:08
%S A224362 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 A224362 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 A224362 6,3,1,4,2,4,6,1,3,4,1,4,3,3,4,5,2,3,4
%N A224362 Number of partitions of n into a prime and a triangular number.
%C A224362 Indices of zeros: 0 followed by A076768.
%H A224362 T. D. Noe, <a href="/A224362/b224362.txt">Table of n, a(n) for n = 0..10000</a>
%F A224362 G.f.: (Sum_{i>=0} x^(i*(i+1)/2))*(Sum_{j>=1} x^prime(j)). - _Ilya Gutkovskiy_, Feb 07 2017
%t A224362 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 A224362 (Python)
%o A224362 import math
%o A224362 primes = [2, 3]
%o A224362 def isprime(k):
%o A224362   for p in primes:
%o A224362     if k%p==0:  return 0
%o A224362   primes.append(k)
%o A224362   return 1
%o A224362 def rootTriangular(a):
%o A224362     sr = 2**(int(math.log(a,2))+2)
%o A224362     while a < sr*(sr+1)//2:
%o A224362           sr>>=1
%o A224362     b = sr>>1
%o A224362     while b:
%o A224362       s = sr+b
%o A224362       if a >= s*(s+1)//2:
%o A224362         sr = s
%o A224362       b>>=1
%o A224362     return sr
%o A224362 for i in range(1<<10):
%o A224362     k = 0
%o A224362     for p in primes:
%o A224362       if i <= p:  continue
%o A224362       r = rootTriangular(i - p)
%o A224362       if r*(r+1)//2 == i-p:  k+=1
%o A224362     if i>1:
%o A224362       if i<=3:  k += 1
%o A224362       else:     k += isprime(i)
%o A224362     print(k, end=', ')
%Y A224362 Cf. A000040, A000217, A076768, A101182.
%K A224362 nonn
%O A224362 0,4
%A A224362 _Alex Ratushnyak_, Apr 04 2013