OFFSET
1,16
COMMENTS
Number of distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(16) = 2; the two rectangles are 3 X 13 and 5 X 11. - Wesley Ivan Hurt, Oct 29 2017
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
FORMULA
G.f.: Sum_{j>0} Sum_{i=1..j-1} x^(p(i)+p(j)), where p(k) is the k-th prime.
G.f.: A(x)^2/2 - A(x^2)/2 where A(x) = Sum_{p in primes} x^p. - Geoffrey Critzer, Nov 21 2012
a(n) = [x^n*y^2] Product_{i>=1} (1+x^prime(i)*y). - Alois P. Heinz, Nov 22 2012
EXAMPLE
a(24) = 3 because we have [19,5], [17,7] and [13,11].
MAPLE
g:=sum(sum(x^(ithprime(i)+ithprime(j)), i=1..j-1), j=1..35): gser:=series(g, x=0, 130): seq(coeff(gser, x, n), n=1..125);
# alternative
A117929 := proc(n)
local a, i, p ;
a := 0 ;
p := 2 ;
for i from 1 do
if 2*p >= n then
return a;
end if;
if isprime(n-p) then
a := a+1 ;
end if;
p := nextprime(p) ;
end do:
end proc:
seq(A117929(n), n=1..80) ; # R. J. Mathar, Oct 01 2021
MATHEMATICA
l = {}; For[n = 1, n <= 1000, n++, c = 0; For[k = 1, Prime[k] < n/2, k++, If[PrimeQ[n - Prime[k]], c = c + 1] ]; AppendTo[l, c] ] l (* Jake Foster, Oct 27 2008 *)
Table[Count[IntegerPartitions[n, {2}], _?(AllTrue[#, PrimeQ]&&#[[1]]!= #[[2]] &)], {n, 120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 26 2020 *)
PROG
(PARI) a(n)=my(s); forprime(p=2, (n-1)\2, s+=isprime(n-p)); s \\ Charles R Greathouse IV, Feb 26 2014
(Python)
from sympy import sieve
from collections import Counter
from itertools import combinations
def aupton(max):
sieve.extend(max)
a = Counter(c[0]+c[1] for c in combinations(sieve._list, 2))
return [a[n] for n in range(1, max+1)]
print(aupton(105)) # Michael S. Branicky, Feb 16 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Apr 03 2006
STATUS
approved