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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 Zak Seidov, Table of n, a(n) for n = 1..10000 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 Cf. A001248, A062772, A054271, A133517, A133518, A133519, A133520, A133521, A133522, A001223. Sequence in context: A205969 A326771 A049782 * A084290 A062011 A225520 Adjacent sequences: A091663 A091664 A091665 * A091667 A091668 A091669 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.)