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

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, 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, 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

Offset: 0

Author

Ilya Gutkovskiy, Mar 18 2017

Keywords

• nonn

Comments

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

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Semiprime
• Eric Weisstein's World of Mathematics, Squarefree
• Eric Weisstein's World of Mathematics, Twin Primes

Crossrefs

Cf. A001097, A006881, A098983, A100949, A282192, A282318.

Programs

Maple

N:= 200: # to get a(0) to a(N)
V:= Vector(N):
Primes:= select(isprime,[2,seq(i,i=3..N+2)]):
PS:= convert(Primes,set);
Twins:= PS intersect map(`-`,PS,2):
Twins:= Twins union map(`+`,Twins,2):
Twins:= sort(convert(Twins,list)):
for i from 1 to nops(Twins) do
  for j from 1 to nops(Primes) while Twins[i]+2*Primes[j] <= N do
    for k from 1 to j-1 do
      v:= Twins[i]+Primes[k]*Primes[j];
      if v > N then break fi;
      V[v]:= V[v]+1;
od od od:
0, seq(V[i],i=1..N); # Robert Israel, Mar 29 2017

Mathematica

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]

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

Formula

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