A337121 a(n) is the number of ways the n-th prime number prime(n) can be represented as sum of two smaller odd prime numbers p1, p2 with prime(n) > p1 > (p2 minus the maximum odd prime factor of (p1-p2)).

1, 1, 2, 3, 2, 2, 4, 3, 3, 5, 3, 6, 4, 4, 4, 7, 4, 4, 5, 6, 6, 6, 9, 7, 8, 8, 7, 7, 6, 11, 4, 11, 9, 7, 8, 9, 7, 13, 12, 6, 10, 15, 10, 9, 7, 13, 13, 11, 13, 10, 15, 10, 13, 14, 11, 13, 13, 12, 14, 17, 13, 13, 19, 9, 14, 19, 12, 8, 14, 22, 17, 14, 13, 16, 9, 15

Offset: 4

Author

Lei Zhou, Aug 17 2020

Keywords

• nonn

Comments

This sequence counts the cases such that prime(n) = p1 + p2 - MaxOddPrimeFactor(p1-p2), where MaxOddPrimeFactor(m) is defined as the maximum odd prime factor of the positive integer m. If there is no odd prime factor of m, MaxOddPrimeFactor(m) is defined as 1.

Conjecture: a(n) > 0 when n >= 4.

Some nonprime odd numbers, like 27, cannot be partitioned into the form of p1 + p2 - MaxOddPrimeFactor(p1-p2).

Examples

			When n=4, prime(4)=7, MaxOddPrimeFactor(5-3)=1, 7=5+3-1. This is the only case, so a(4)=1.
When n=5, prime(5)=11, MaxOddPrimeFactor(7-5)=1, 11=7+5-1. This is the only case, so a(5)=1.
When n=6, prime(6)=13, MaxOddPrimeFactor(11-3)=1, 13=11+3-1; and MaxOddPrimeFactor(11-5)=3, 13=11+5-3. Two cases found, so a(6)=2.
		

Programs

Mathematica

MaxOddPrimeFactor[m_] :=
Module[{factors, l, res}, factors = FactorInteger[m];
  l = Length[factors]; res = factors[[l, 1]]; If[res == 2, res = 1];
  res]
Table[p = Prime[n]; p1 = NextPrime[p/2, -1]; ct = 0;
While[p1 = NextPrime[p1]; p1 < p, p2 = NextPrime[p - p1, -1];
  While[p2 = NextPrime[p2]; p2 < p1,
   If[p == (p1 + p2 - MaxOddPrimeFactor[p1 - p2]), ct++]]]; ct, {n, 4,
   79}]