cp's OEIS Frontend

A116981 Number of distinct representations of 8n^3 as the sum of two primes.

%I A116981 #13 Oct 11 2019 16:19:27
%S A116981 1,5,13,11,28,53,50,53,135,106,116,253,165,229,568,244,313,656,381,
%T A116981 575,1123,600,612,1297,956,871,1735,1130,1102,3025,1288,1314,3169,
%U A116981 1620,2671,3582,1954,2149,4729,3064,2513,6244,2822,3276,8242,3450,3590,7305,4598,5402,9028,4825,4809,9886,7552,6446
%N A116981 Number of distinct representations of 8n^3 as the sum of two primes.
%D A116981 H. Halberstam and H. E. Richert, "Sieve methods", Academic Press, London, New York, San Francisco, 1974.
%F A116981 a(n) = #{p(i) + p(j) = (2n)^3 for p(k) = A000040(k) and i >= j}. a(n) = #{p(i) + p(j) = A016743(n) for p(k) = A000040(k) and i >= j}.
%e A116981 a(1) = 1 because (2*1)^3 = 8 = 3 + 5 uniquely.
%e A116981 a(2) = 5 because (2*2)^3 = 64 = 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.
%e A116981 a(3) = 13 because (2*3)^3 = 216 = 5 + 211 = 17 + 199 = 19 + 197 = 23 + 193 = 37 + 179 = 43 + 173 = 53 + 163 = 59 + 157 = 67 + 149 = 79 + 137 = 89 + 127 = 103 + 113 = 107 + 109.
%p A116981 a:=proc(n) local ct,j: ct:=0: for j from 1 to prevprime((2*n)^3) do if isprime((2*n)^3-ithprime(j))=true then ct:=ct+1 else ct:=ct fi od: ct/2: end: seq(a(n),n=1..40); # execution takes hours - _Emeric Deutsch_, Apr 17 2006
%p A116981 # Faster alternative
%p A116981 N:= 100: # to get a(1)..a(N)
%p A116981 V:= Vector(8*N^3,datatype=float[8]):
%p A116981 P:= select(isprime, [2,seq(i,i=3..8*N^3,2)]):
%p A116981 V[P]:= 1:
%p A116981 C:= SignalProcessing:-Convolution(V,V):
%p A116981 seq(round(C[8*n^3-1])/2,n=1..N); # _Robert Israel_, Jan 24 2018
%t A116981 Table[Count[IntegerPartitions[8*n^3,{2}],_?(AllTrue[#,PrimeQ]&)],{n,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (*The program will take a long time to run. *) (* _Harvey P. Dale_, Oct 11 2019 *)
%Y A116981 Cf. A000040, A016742, A016743.
%K A116981 easy,nonn
%O A116981 1,2
%A A116981 _Jonathan Vos Post_, Apr 01 2006
%E A116981 Corrected and extended by _Emeric Deutsch_, Apr 17 2006
%E A116981 More terms from _Robert Israel_, Jan 24 2018