A062772 -id:A062772 - cp's OEIS frontend

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

Dimitris Valianatos, Dec 11 2020

sign

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.

			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.

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

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

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

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

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

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

A091666 Difference between prime(n)^2 and the next prime.

Original entry on oeis.org

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

Pierre CAMI, Jan 27 2004

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]

			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.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A001248, A062772, A054271, A133517, A133518, A133519, A133520, A133521, A133522, A001223.

NextPrime[#^2]-#^2&/@Prime[Range[74]] (* Zak Seidov, Apr 18 2012 *)

a(n) = my(x=prime(n)^2); nextprime(x)-x; \\ Michel Marcus, Oct 07 2023

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} log(prime(n))) = 2. - Alain Rocchelli, Oct 04 2023

A062773 Index of the smallest prime which follows square of n-th prime.

Original entry on oeis.org

3, 5, 10, 16, 31, 40, 62, 73, 100, 147, 163, 220, 264, 284, 330, 410, 488, 520, 610, 676, 706, 812, 887, 1001, 1164, 1253, 1295, 1382, 1424, 1524, 1878, 1977, 2142, 2191, 2490, 2548, 2730, 2916, 3044, 3242, 3437, 3513, 3869, 3946, 4090, 4165, 4628

Offset: 1

Labos Elemer, Jul 18 2001

nonn

			100th prime, 541 immediately follows 529, square of 9th prime, a(9)=100.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A054270, A054271, A000879.

PrimePi[NextPrime[#]]&/@(Prime[Range[50]]^2) (* Harvey P. Dale, Apr 12 2023 *)

a(n) = { primepi(prime(n)^2) + 1 } \\ Harry J. Smith, Aug 10 2009

a(n) = pi( nextprime( prime(n)^2 ) ).

a(n) = A000720(A062772(n)). - Michel Marcus, Jun 24 2014

Offset changed from 0 to 1 by Harry J. Smith, Aug 10 2009

cp's OEIS Frontend

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

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A091666 Difference between prime(n)^2 and the next prime.

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A062773 Index of the smallest prime which follows square of n-th prime.

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