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A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881). 1

%I

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

%C Conjecture: a(n) > 0 for all n > 30.

%H Robert Israel, <a href="/A283929/b283929.txt">Table of n, a(n) for n = 0..10000</a>

%H Ilya Gutkovskiy, <a href="/A283929/a283929.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semiprime.html">Semiprime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>

%F G.f.: (Sum_{k>=1} x^A001097(k))*(Sum_{k>=1} x^A006881(k)).

%e a(17) = 3 because we have [14, 3], [11, 6] and [10, 7].

%p N:= 200: # to get a(0) to a(N)

%p V:= Vector(N):

%p Primes:= select(isprime,[2,seq(i,i=3..N+2)]):

%p PS:= convert(Primes,set);

%p Twins:= PS intersect map(`-`,PS,2):

%p Twins:= Twins union map(`+`,Twins,2):

%p Twins:= sort(convert(Twins,list)):

%p for i from 1 to nops(Twins) do

%p for j from 1 to nops(Primes) while Twins[i]+2*Primes[j] <= N do

%p for k from 1 to j-1 do

%p v:= Twins[i]+Primes[k]*Primes[j];

%p if v > N then break fi;

%p V[v]:= V[v]+1;

%p od od od:

%p 0, seq(V[i],i=1..N); # _Robert Israel_, Mar 29 2017

%t 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 (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 Cf. A001097, A006881, A098983, A100949, A282192, A282318.

%K nonn

%O 0,14

%A _Ilya Gutkovskiy_, Mar 18 2017

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Last modified February 24 23:02 EST 2018. Contains 299629 sequences. (Running on oeis4.)