A224362 Number of partitions of n into a prime and a triangular number.
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, 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, 6, 3, 1, 4, 2, 4, 6, 1, 3, 4, 1, 4, 3, 3, 4, 5, 2, 3, 4
Offset: 0
Keywords
Links
- T. D. Noe, Table of n, a(n) for n = 0..10000
Programs
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Mathematica
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 *) -
Python
import math primes = [2, 3] def isprime(k): for p in primes: if k%p==0: return 0 primes.append(k) return 1 def rootTriangular(a): sr = 2**(int(math.log(a,2))+2) while a < sr*(sr+1)//2: sr>>=1 b = sr>>1 while b: s = sr+b if a >= s*(s+1)//2: sr = s b>>=1 return sr for i in range(1<<10): k = 0 for p in primes: if i <= p: continue r = rootTriangular(i - p) if r*(r+1)//2 == i-p: k+=1 if i>1: if i<=3: k += 1 else: k += isprime(i) print(k, end=', ')
Formula
G.f.: (Sum_{i>=0} x^(i*(i+1)/2))*(Sum_{j>=1} x^prime(j)). - Ilya Gutkovskiy, Feb 07 2017
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