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)).

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

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

Lei Zhou, Table of n, a(n) for n = 1..219

A. Granville, T. Agoh, 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.

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

Cf. A210529

Sequence in context: A010772 A199732 A293238 * A210529 A151785 A093564

Adjacent sequences: A210705 A210706 A210707 * A210709 A210710 A210711

Keyword

nonn

Author

Lei Zhou, Jan 29 2013

Status

approved