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A117929
Number of partitions of n into 2 distinct primes.
19
0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1, 2, 0, 2, 0, 3, 1, 2, 1, 3, 0, 4, 0, 1, 1, 3, 0, 4, 1, 3, 1, 3, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 3, 0, 6, 1, 2, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 4, 1, 5, 0, 7, 0, 4, 1, 4, 0, 8, 1, 4, 0, 4, 0, 9, 1, 4, 0, 4, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1
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
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
a(n) = Sum_{i=2..floor((n-1)/2)} A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017
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
Cf. A010051, A045917, A061358, A073610, A166081 (positions of 0), A077914 (positions of 2), A080862 (positions of 6).
Column k=2 of A219180. - Alois P. Heinz, Nov 13 2012
Sequence in context: A178687 A325538 A238417 * A306439 A107455 A039701
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Apr 03 2006
STATUS
approved