|
%I #5 Mar 30 2012 18:40:36
%S 50,200,260,290,370,530,578,610,650,740,884,962,1060,1170,1300,1370,
%T 1460,1508,1530,1690
%N Number of distinct representations of n as the sum of two nonzero squares nontrivially divides the number of distinct representations of n as the sum of two primes.
%C "Nontrivially" meaning the number of distinct representations of n as the sum of two nonzero squares is at least 2.
%F Numbers n such that #{a^2 + b^2 = n and a>0 and b>0 and a>= b} > 1 and #{a^2 + b^2 = n and a>0 and b>0 and a>= b} | #{p(i) + p(j) = n and i >= j where p(k) = A000040(k)}.
%e a(1) = 50 because 50 = 1^2 + 49^2 = 5^2 + 5^2 (2 distinct ways as sum of nonzero squares) and 50 = 3 + 47 = 7 + 43 = 13 + 37 = 19 + 31 (4 distinct ways as sum of two primes) and 2 | 4.
%e a(2) = 200 because 200 = 2^2 + 14^2 = 10^2 + 10^2 (2 distinct ways as sum of nonzero squares) and 200 = 3 + 197 = 7 + 193 = 19 + 181 = 37 + 163 = 43 + 157 = 61 + 139 = 73 + 127 = 97 + 103, (8 distinct ways as sum of two primes) and 2 | 8.
%e a(3) = 260 because (2 distinct ways as sum of nonzero squares) divides (10 distinct ways as sum of two primes).
%e a(4) = 290 because (2 distinct ways as sum of nonzero squares) divides (10 distinct ways as sum of two primes).
%e a(5) = 370 because (2 distinct ways as sum of nonzero squares) divides (14 distinct ways as sum of two primes).
%e a(6) = 530 because (2 distinct ways as sum of nonzero squares) divides (14 distinct ways as sum of two primes).
%e a(7) = 578 because (2 distinct ways as sum of nonzero squares) divides (12 distinct ways as sum of two primes).
%e a(8) = 610 because (2 distinct ways as sum of nonzero squares) divides (20 distinct ways as sum of two primes).
%e a(9) = 650 because (3 distinct ways as sum of nonzero squares) divides (21 distinct ways as sum of two primes).
%e a(10) = 740 because (2 distinct ways as sum of nonzero squares) divides (18 distinct ways as sum of two primes).
%e 1300 is in the sequence because (3 distinct ways as sum of nonzero squares) divides (33 distinct ways as sum of two primes).
%Y Cf. A000040, A025284-A025293.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Apr 02 2006
%E More terms from Nate Falkenstein (njf127(AT)psu.edu), Apr 25 2006
| 