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

1,1

Comments

"Nontrivially" meaning the number of distinct representations of n as the sum of two nonzero squares is at least 2.

Links

Table of n, a(n) for n=1..20.

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)}.

Example

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).

Crossrefs

Cf. A000040, A025284-A025293.

Sequence in context: A180293 A031692 A173141 * A334808 A273293 A097371

Adjacent sequences: A115589 A115590 A115591 * A115593 A115594 A115595

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 02 2006

Extensions

More terms from Nate Falkenstein (njf127(AT)psu.edu), Apr 25 2006

Status

approved