|
| |
|
|
A216501
|
|
Let S_k = {x^2+k*y^2: x,y positive integers}. How many out of S_1, S_2, S_3, S_7 does n belong to?
|
|
7
|
|
|
|
0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 0, 2, 3, 1, 1, 1, 2, 0, 3, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 0, 2, 0, 1, 2, 1, 1, 3, 2, 0, 0, 1, 3, 3, 1, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 1, 3, 1, 2, 2, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 1, 2, 3, 1, 0, 2, 2, 1, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,8
|
|
|
COMMENTS
|
"If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor P (of C) raised to an odd power is of the form c^2 + kd^2, for some integers c & d."
This statement is only true for k = 1, 2, 3. For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
A number can be written as a^2 + b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
A number can be written as a^2 + 2b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
A number can be written as a^2 + 3b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
A number can be written as a^2 + 7b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power and the exponent of 2 is not 1.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
PROG
|
(PARI) for(n=1, 100, sol=0; for(x=1, 100, if(issquare(n-x*x)&&n-x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-2*x*x)&&n-2*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-3*x*x)&&n-3*x*x>0, sol++; break)); for(x=1, 100, if(issquare(n-7*x*x)&&n-7*x*x>0, sol++; break)); print1(sol", ")) /* V. Raman, Oct 16 2012 */
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
