A115592 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.
50, 200, 260, 290, 370, 530, 578, 610, 650, 740, 884, 962, 1060, 1170, 1300, 1370, 1460, 1508, 1530, 1690
Offset: 1
Examples
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. 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. a(3) = 260 because (2 distinct ways as sum of nonzero squares) divides (10 distinct ways as sum of two primes). a(4) = 290 because (2 distinct ways as sum of nonzero squares) divides (10 distinct ways as sum of two primes). a(5) = 370 because (2 distinct ways as sum of nonzero squares) divides (14 distinct ways as sum of two primes). a(6) = 530 because (2 distinct ways as sum of nonzero squares) divides (14 distinct ways as sum of two primes). a(7) = 578 because (2 distinct ways as sum of nonzero squares) divides (12 distinct ways as sum of two primes). a(8) = 610 because (2 distinct ways as sum of nonzero squares) divides (20 distinct ways as sum of two primes). a(9) = 650 because (3 distinct ways as sum of nonzero squares) divides (21 distinct ways as sum of two primes). a(10) = 740 because (2 distinct ways as sum of nonzero squares) divides (18 distinct ways as sum of two primes). 1300 is in the sequence because (3 distinct ways as sum of nonzero squares) divides (33 distinct ways as sum of two primes).
Formula
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)}.
Extensions
More terms from Nate Falkenstein (njf127(AT)psu.edu), Apr 25 2006
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