|
%I #22 May 09 2019 05:04:31
%S 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,
%T 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,
%U 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
%N Number of partitions of n into a prime and a semiprime.
%C 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
%D Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.
%H T. D. Noe, <a href="/A100949/b100949.txt">Table of n, a(n) for n = 1..10000</a>
%F A100951(n) <= A100950(n) <= a(n) <= min(A000720(n), A072000(n)).
%F a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A064911(n-i) + A010051(n-i) * A064911(i). - _Wesley Ivan Hurt_, May 02 2019
%e a(21) = #{7+2*7, 11+2*5, 17+2*2} = 3.
%t Table[Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}],{n,110}] (* _Harvey P. Dale_, Mar 25 2018 *)
%o (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
%o (Haskell)
%o a100949 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a001358_list
%o -- _Reinhard Zumkeller_, Jun 26 2013
%Y Cf. A061358, A000040, A001358.
%Y Cf. A010051.
%K nonn,easy
%O 1,9
%A _Reinhard Zumkeller_, Nov 23 2004
|
