|
|
A210708
|
|
a(n) is the smallest positive number coprime to prime(n) such that |a(n)^2-prime(n)^2| is divisible by all primes less than sqrt(prime(n)).
|
|
2
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 7, 7, 17, 11, 19, 17, 1, 17, 19, 13, 19, 13, 11, 23, 23, 11, 13, 83, 89, 17, 29, 61, 179, 283, 233, 13, 1213, 1999, 391, 719, 1523, 2507, 529, 1219, 2533, 1943, 541, 1223, 421, 1319, 1681, 653, 1277, 1369, 821, 563, 1721
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,12
|
|
COMMENTS
|
Suppose a = a(n)+prime(n), b = |a(n)-prime(n)|, when a(n) > prime(n), prime(n) = (a - b)/2, and gcd(a,b) = 2 since a(n) and prime(n) are coprime. When a*b = |a(n)^2-prime(n)^2|, (a - b)/2 is a primality proof of prime(n) since the prime factors of a and b contains all prime numbers less than sqrt(prime(n)) and gcd(a,b) = 2. - corrected by Eric M. Schmidt, Feb 02 2013
When a(n) is prime, a(n)=A210529(n); when a(n) is composite, a(n) does not have any prime factors less than sqrt(prime(n)).
If the primes less than sqrt(prime(n)) are p_1, ..., p_r, then k = |prime(n) - p_1*...*p_r| is coprime to prime(n), and k^2 - prime(n)^2 is divisible by all of p_1, ..., p_r. So the sequence is defined for all positive integers n. - Eric M. Schmidt, Feb 02 2013
|
|
LINKS
|
R. K. Guy, C. B. Lacampagne and J. L. Selfridge, Primes at a glance, Math. Comp. 48 (1987), 183-202.
|
|
MATHEMATICA
|
Table[p = Prime[n]; t = Product[Prime[k], {k, 1, PrimePi[NextPrime[Floor[Sqrt[p]] + 1, -1]]}]; p1 = 1; While[r = Abs[p^2 - p1^2]; (r == 0) || (Mod[r, t] != 0), p1++]; p1, {n, 1, 60}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|