

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
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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



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



