login
A091666
Difference between prime(n)^2 and the next prime.
10
1, 2, 4, 4, 6, 4, 4, 6, 12, 12, 6, 4, 12, 12, 4, 10, 10, 6, 4, 10, 4, 6, 10, 6, 4, 10, 4, 18, 6, 12, 10, 6, 4, 12, 28, 6, 10, 4, 4, 18, 10, 10, 12, 4, 12, 6, 10, 10, 10, 12, 4, 10, 18, 28, 18, 22, 6, 12, 4, 16, 18, 4, 4, 10, 4, 4, 6, 22, 4, 42, 24, 22, 10, 4
OFFSET
1,2
COMMENTS
Conjecturally, a(n) << log^2 n (with constant around 8/e^gamma in the supremum). [Charles R Greathouse IV, Dec 27 2011]
Except for a(2)=2, there are no terms = 2 mod 6 (as p^2+2 = 0 mod 3 for primes p > 3). Also, only 1 and 2 appear once while all other terms may appear (infinitely) many times. [Zak Seidov, Apr 18 2012]
FORMULA
Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} log(prime(n))) = 2. - Alain Rocchelli, Oct 04 2023
EXAMPLE
prime(3)=5, 5*5=25 for k=4 25+4=29 prime, k=4 is the least k with prime(3)^2 + k prime.
MATHEMATICA
NextPrime[#^2]-#^2&/@Prime[Range[74]] (* Zak Seidov, Apr 18 2012 *)
PROG
(PARI) a(n) = my(x=prime(n)^2); nextprime(x)-x; \\ Michel Marcus, Oct 07 2023
KEYWORD
easy,nonn
AUTHOR
Pierre CAMI, Jan 27 2004
STATUS
approved