A339640 a(n) = (A062772(n) + A054270(n)) / 2 - A001248(n).

0, 0, 1, 1, -1, 1, -1, 2, 3, 5, -1, 1, 0, 5, 1, 2, -1, 2, -1, 4, -1, -3, 2, 2, -1, 1, 1, 8, -4, 3, 4, 2, -4, 5, 10, -4, -4, -2, -1, 8, -1, -1, 5, -1, 3, -7, 4, 4, 1, 2, 1, 4, 5, 8, 8, 8, -1, 2, -4, -2, 3, 1, -8, -4, 1, -1, -4, 10, -2, 15, 8, 10, 2

Offset: 1

Author

Dimitris Valianatos, Dec 11 2020

Keywords

• sign

Comments

Conjecture: The partial sums of this sequence are greater than or equal to zero. This means that the squares of the prime numbers are smaller than the average of the previous and the next prime number most of the time.

Examples

			For n = 10 prime(10)^2 = 29^2 = 841. The previous prime of 841 is 839 and the next 853. The average of 839 and 853 is (839 + 853)/2 = 846. So a(10) = 846 - 841 = 5.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001248, A054270, A062772, A123993.

Programs

Maple

f:= p -> (nextprime(p^2) + prevprime(p^2))/2 - p^2:
map(f, [seq(ithprime(i),i=1..100)]); # Robert Israel, Nov 24 2024

Mathematica

Array[(Total@ NextPrime[#, {-1, 1}])/2 - # &[Prime[#]^2] &, 73] (* Michael De Vlieger, Dec 11 2020 *)

PARI

forprime(n = 2, 370, print1((nextprime(n^2) + precprime(n^2)) / 2 - n^2", "))

Formula

a(n) = (nextprime(prime(n)^2) + precprime(prime(n)^2)) / 2 - prime(n)^2.