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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
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 (list; graph; refs; listen; history; text; internal format)
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: A106844 A125989 A125924 * A082513 A187495 A187496

Adjacent sequences:  A283926 A283927 A283928 * A283930 A283931 A283932

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 18 2017

STATUS

approved

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Last modified February 22 18:00 EST 2018. Contains 299469 sequences. (Running on oeis4.)