|
| |
|
|
A210529
|
|
a(n) is 1 or the smallest prime that makes |a(n)^2 - prime(n)^2| divisible by all primes smaller than sqrt(prime(n)).
|
|
3
|
|
|
|
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, 2029, 719, 1523, 2927, 2089, 3221, 5657, 6857, 541, 1223, 421, 1319, 3709, 653, 1277, 3371, 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. When a*b = |a(n)^2 - prime(n)^2|, (a - b)/2 is a primality proof of prime(n) since the list of prime factors of a and b contains all prime numbers smaller than sqrt(prime(n)) and gcd(a,b) = 2. - corrected by Eric M. Schmidt, Feb 02 2013
Conjecture: a(n) is defined for all positive integers n.
|
|
|
LINKS
|
R. K. Guy, C. B. Lacampagne and J. L. Selfridge, Primes at a glance, Math. Comp. 48 (1987), 183-202.
|
|
|
EXAMPLE
|
n = 1, prime(1) = 2, the set of prime numbers smaller than sqrt(2) is {}, so a(1) = 1.
n = 11, prime(11) = 31, the set of prime numbers smaller than sqrt(31) is {2, 3, 5}, 961 - 1 = 960 is divisible by 2*3*5, so a(11) = 1.
n = 12, prime(12) = 37, the set of prime numbers smaller than sqrt(37) is {2, 3, 5}, 37^2 - 7^2 = 1320 is divisible by 2*3*5, so a(12) = 7.
|
|
|
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 = NextPrime[p1]]; p1, {n, 1, 60}]
|
|
|
PROG
|
(PARI) primorial(n)=vecprod(primes(n));
a(n)=if (n<=3, 1, my(p=prime(n), P=primorial(sqrtint(p)), p2=p^2); if(p2%P==1, return(1)); forprime(q=2, , if((q^2-p^2)%P==0&&p!=q, return(q)))) \\ Charles R Greathouse IV, Mar 01 2014; edited by Michel Marcus, Oct 22 2023
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,hard,nice
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
