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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 (list; graph; refs; listen; history; text; internal format)
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]
LINKS
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
CROSSREFS
Sequence in context: A205969 A326771 A049782 * A084290 A062011 A225520
KEYWORD
easy,nonn
AUTHOR
Pierre CAMI, Jan 27 2004
STATUS
approved

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Last modified April 17 11:20 EDT 2024. Contains 371763 sequences. (Running on oeis4.)