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

0,4

Comments

Indices of zeros: 0 followed by A076768.

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Formula

G.f.: (Sum_{i>=0} x^(i*(i+1)/2))*(Sum_{j>=1} x^prime(j)). - Ilya Gutkovskiy, Feb 07 2017

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 *)

Prog

(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=', ')

Crossrefs

Cf. A000040, A000217, A076768, A101182.

Sequence in context: A263643 A336534 A271852 * A350560 A050362 A095686

Adjacent sequences: A224359 A224360 A224361 * A224363 A224364 A224365

Keyword

nonn

Author

Alex Ratushnyak, Apr 04 2013

Status

approved