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
Lei Zhou and Charles R Greathouse IV, Table of n, a(n) for n = 1..270 (first 165 terms from Zhou)
T. Agoh, A. Granville, and P. Erdős, Primes at a (somewhat lengthy) glance, American Mathematical Monthly, 104(10):943-945, December 1997.
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
Lei Zhou, Jan 27 2013
STATUS
approved