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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

Lei Zhou and Charles R Greathouse IV, Table of n, a(n) for n = 1..270 (first 165 terms from Zhou)

A. Granville, T. Agoh, and P. Erdos, 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) a(n)=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

CROSSREFS

Sequence in context: A199732 A293238 A210708 * A151785 A093564 A081776

Adjacent sequences:  A210526 A210527 A210528 * A210530 A210531 A210532

KEYWORD

nonn,hard,nice

AUTHOR

Lei Zhou, Jan 27 2013

STATUS

approved

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Last modified August 5 02:34 EDT 2021. Contains 346457 sequences. (Running on oeis4.)