A100949 Number of partitions of n into a prime and a semiprime.

0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 2, 5, 1, 2, 2, 3, 2, 4, 2, 3, 3, 5, 5, 4, 1, 2, 4, 5, 2, 4, 3, 5, 6, 4, 5, 6, 3, 4, 5, 6, 5, 4, 3, 4, 4, 8, 7, 6, 4, 3, 7, 8, 6, 4, 4, 3, 10, 7, 6, 7, 4, 6, 10, 7, 6, 5, 6, 4, 7, 8, 9, 7, 5, 6, 9, 8, 9, 4, 5, 7, 8, 9, 11, 8, 4, 4, 11, 12, 10, 6, 10, 7, 13, 9, 9, 6

Offset: 1

Author

Reinhard Zumkeller, Nov 23 2004

Keywords

• nonn
• easy

Comments

Marnell conjectures that a(n) > 0 for n > 10 after analyzing "many thousands of whole numbers". I find no exceptions below 100 million. - Charles R Greathouse IV, May 04 2010

Examples

			a(21) = #{7+2*7, 11+2*5, 17+2*2} = 3.
		

References

• Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A061358, A000040, A001358.

Cf. A010051.

Programs

Haskell

a100949 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a001358_list
-- Reinhard Zumkeller, Jun 26 2013

Mathematica

Table[Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}],{n,110}] (* Harvey P. Dale, Mar 25 2018 *)

PARI

list(lim)=my(p=primes(primepi(lim)),sp=select(n->bigomega(n)==2, vector(lim\1,i,i)),x=O('x^(lim\1+1))+'x); concat([0,0,0,0,0], Vec(sum(i=1,#p,x^p[i])*sum(i=1,#sp,x^sp[i]))) \\ Charles R Greathouse IV, Jun 14 2013

Formula

A100951(n) <= A100950(n) <= a(n) <= min(A000720(n), A072000(n)).

a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A064911(n-i) + A010051(n-i) * A064911(i). - Wesley Ivan Hurt, May 02 2019