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Number of ordered ways of writing n as the sum of two semiprimes.
3

%I #24 Jan 07 2024 00:59:47

%S 0,0,0,0,0,0,0,1,0,2,0,1,2,2,2,2,0,3,4,3,2,0,2,4,4,2,2,3,4,5,6,4,0,2,

%T 6,6,4,2,6,6,4,5,8,7,4,2,8,10,6,5,2,5,6,4,10,6,4,4,10,12,12,2,6,10,6,

%U 7,8,9,6,5,12,14,10,6,6,7,8,7,10,10,6,4,14,14

%N Number of ordered ways of writing n as the sum of two semiprimes.

%C Conjecture: Only the integers 1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22, 33 (A072966) cannot be partitioned into a set of two semiprimes.

%H Robert Israel, <a href="/A199331/b199331.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A064911. - _Wesley Ivan Hurt_, Jan 07 2024

%p sp:= select(t -> numtheory:-bigomega(t)=2, [$1..100]):

%p G:= expand(add(x^t,t=sp)^2):

%p seq(coeff(G,x,i),i=1..100); # _Robert Israel_, Nov 24 2020

%t mx=200; semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; t = Select[ Range@ mx, semiPrimeQ]; s = Sort[Plus @@@ Tuples[t, 2]]; Transpose[ Tally@ Join[ Range@ mx, s]][[2]] - 1

%Y Cf. A001358, A035026, A064911, A072966, A073610.

%K easy,nonn

%O 1,10

%A _Robert G. Wilson v_, Nov 05 2011

%E Definition clarified by _Robert Israel_, Nov 24 2020