A224362 - OEIS

%I #17 Nov 14 2024 10:58:08

%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, 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(1<<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(k, end=', ')

%Y Cf. A000040, A000217, A076768, A101182.

%K nonn

%O 0,4

%A _Alex Ratushnyak_, Apr 04 2013