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A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881).

%I A283929 #13 Feb 16 2025 08:33:43
%S A283929 0,0,0,0,0,0,0,0,0,1,0,1,0,2,0,1,0,3,1,2,1,2,1,2,1,3,2,3,2,3,0,2,2,3,
%T A283929 2,2,1,3,3,4,3,4,2,3,3,4,4,2,1,3,3,5,4,4,2,3,3,4,4,1,2,1,5,4,5,6,2,4,
%U A283929 5,5,4,2,3,2,5,5,6,5,2,4,5,5,6,2,3,4,4,6,5,4,3,3,5,6,8,3,7,4,9,6,6,3,3,3,5,6,7,4,5,3,5
%N A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881).
%C A283929 Conjecture: a(n) > 0 for all n > 30.
%H A283929 Robert Israel, <a href="/A283929/b283929.txt">Table of n, a(n) for n = 0..10000</a>
%H A283929 Ilya Gutkovskiy, <a href="/A283929/a283929.pdf">Extended graphical example</a>
%H A283929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%H A283929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A283929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%F A283929 G.f.: (Sum_{k>=1} x^A001097(k))*(Sum_{k>=1} x^A006881(k)).
%e A283929 a(17) = 3 because we have [14, 3], [11, 6] and [10, 7].
%p A283929 N:= 200: # to get a(0) to a(N)
%p A283929 V:= Vector(N):
%p A283929 Primes:= select(isprime,[2,seq(i,i=3..N+2)]):
%p A283929 PS:= convert(Primes,set);
%p A283929 Twins:= PS intersect map(`-`,PS,2):
%p A283929 Twins:= Twins union map(`+`,Twins,2):
%p A283929 Twins:= sort(convert(Twins,list)):
%p A283929 for i from 1 to nops(Twins) do
%p A283929   for j from 1 to nops(Primes) while Twins[i]+2*Primes[j] <= N do
%p A283929     for k from 1 to j-1 do
%p A283929       v:= Twins[i]+Primes[k]*Primes[j];
%p A283929       if v > N then break fi;
%p A283929       V[v]:= V[v]+1;
%p A283929 od od od:
%p A283929 0, seq(V[i],i=1..N); # _Robert Israel_, Mar 29 2017
%t A283929 nmax = 110; CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k, {k, 1, nmax}] Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}], {x, 0, nmax}], x]
%o A283929 (PARI) concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k) * sum(k=2, 110, moebius(k)^2 * floor(2/bigomega(k)) * floor(bigomega(k)/2) * x^k) + O(x^111))) \\ _Indranil Ghosh_, Mar 18 2017
%Y A283929 Cf. A001097, A006881, A098983, A100949, A282192, A282318.
%K A283929 nonn
%O A283929 0,14
%A A283929 _Ilya Gutkovskiy_, Mar 18 2017