A210497 a(n) = 2*prime(n+1) - prime(n).

4, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273, 283

Offset: 1

Author

Marco Piazzalunga, Jan 24 2013

Keywords

• nonn
• easy

Comments

The subsequence of multiples of 3 begins: 9, 15, 15, 21, 21, 27, 33, 45.

The subsequence of primes begins: 7, 43, 73, 163, 179, 223.

Some terms, like a(3)=15 or a(5)=21, are repeated twice, other terms, like a(23)=105, are repeated three times.

Examples

			a(2) = 7 because prime(3) = 5, prime(2) = 3, and 2 * 5 - 3 = 7.
a(3) = 9 because prime(4) = 7, prime(3) = 5, and 2 * 7 - 5 = 9.
a(4) = 15 because prime(5) = 11, prime(4) = 7, and 2 * 11 - 7 = 15.
		

Links

• Bruno Berselli and Zak Seidov, Table of n, a(n) for n = 1..10000 (a(1)-a(1000) from Bruno Berselli).

Crossrefs

Cf. A001223, A062234, A085704 (subsequence).

Programs

Magma

[2*NextPrime(p)-p: p in PrimesUpTo(300)]; // Bruno Berselli, Jan 24 2013

Mathematica

Table[2 Prime[n + 1] - Prime[n], {n, 50}] (* Vincenzo Librandi, May 03 2015 *)
ListConvolve[{2, -1}, Prime[Range[100]]] (* Paolo Xausa, Oct 29 2024 *)

PARI

a(n)=my(p=prime(n));2*nextprime(p+1)-p \\ Charles R Greathouse IV, Jan 24 2013

Python

from sympy import prime, nextprime
def A210497(n): return -(p:=prime(n))+(nextprime(p)<<1) # Chai Wah Wu, Oct 29 2024

Formula

a(n) ~ n log n. - Charles R Greathouse IV, Jan 24 2013