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

0,14

Comments

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

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

Formula

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

Example

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

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]

Prog

(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

Crossrefs

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

Sequence in context: A363904 A125989 A125924 * A316401 A082513 A187495

Adjacent sequences: A283926 A283927 A283928 * A283930 A283931 A283932

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 18 2017

Status

approved