A116981 - OEIS

A116981

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

1, 5, 13, 11, 28, 53, 50, 53, 135, 106, 116, 253, 165, 229, 568, 244, 313, 656, 381, 575, 1123, 600, 612, 1297, 956, 871, 1735, 1130, 1102, 3025, 1288, 1314, 3169, 1620, 2671, 3582, 1954, 2149, 4729, 3064, 2513, 6244, 2822, 3276, 8242, 3450, 3590, 7305, 4598, 5402, 9028, 4825, 4809, 9886, 7552, 6446

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OFFSET

1,2

REFERENCES

H. Halberstam and H. E. Richert, "Sieve methods", Academic Press, London, New York, San Francisco, 1974.

LINKS

Table of n, a(n) for n=1..56.

FORMULA

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}.

EXAMPLE

a(1) = 1 because (2*1)^3 = 8 = 3 + 5 uniquely.

a(2) = 5 because (2*2)^3 = 64 = 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.

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.

MAPLE

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

# Faster alternative

N:= 100: # to get a(1)..a(N)

V:= Vector(8*N^3, datatype=float[8]):

P:= select(isprime, [2, seq(i, i=3..8*N^3, 2)]):

V[P]:= 1:

C:= SignalProcessing:-Convolution(V, V):

seq(round(C[8*n^3-1])/2, n=1..N); # Robert Israel, Jan 24 2018

MATHEMATICA

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 *)

CROSSREFS

Cf. A000040, A016742, A016743.

Sequence in context: A299190 A156682 A089534 * A222756 A094150 A375275

Adjacent sequences: A116978 A116979 A116980 * A116982 A116983 A116984

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Apr 01 2006

EXTENSIONS

Corrected and extended by Emeric Deutsch, Apr 17 2006

More terms from Robert Israel, Jan 24 2018

STATUS

approved