A040976 a(n) = prime(n) - 2.

0, 1, 3, 5, 9, 11, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 65, 69, 71, 77, 81, 87, 95, 99, 101, 105, 107, 111, 125, 129, 135, 137, 147, 149, 155, 161, 165, 171, 177, 179, 189, 191, 195, 197, 209, 221, 225, 227, 231, 237, 239, 249, 255, 261

Offset: 1

Author

Felice Russo

Keywords

• nonn
• nice
• easy

Comments

Numbers k such that k! reduced mod (k+2) is 1. - Benoit Cloitre, Mar 11 2002

The first a(n) numbers starting from 2 are divisible by primes up to prime(n-1). - Lekraj Beedassy, Jun 21 2006

The terms in this sequence are the cumulative sums of distances from one prime to another. For example for the distance from the first to 26th prime, 2 to 101, the cumulative sum of distances is 99, always the last prime, here 101, minus 2. - Enoch Haga, Apr 24 2006

The primes in this sequence are the initial primes of twin prime pairs. - Sebastiao Antonio da Silva, Dec 21 2008

Note that many, but not all, of these numbers satisfy x such that x^(x+1) = 1 mod (x+2). The first exception is 339. - Thomas Ordowski, Nov 27 2013

If this sequence had an infinite number of primes, the twin prime conjecture would follow. Sequence holds all primes in A001359. - John W. Nicholson, Apr 14 2014

From Bernard Schott, Feb 19 2023: (Start)

Equivalently, except for a(1)=0, all terms are odd integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

For each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 2, so this unique AP is (2, 2+d) = (2, prime(m)) with m > 1; so, first examples are (2,3), (2,5), (2,7), (2,11), ... next elements should be respectively: 4, 8, 12, 20, ... that are all composite numbers.

Similar sequence with even common differences d is A360735.

This subsequence of A359408 corresponds to the first case: '2 is prime'; second case corresponding to the even common differences d is A360735. (End)

Examples

			a(13) = 39, because A000040(13) = 41.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• S. A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
• S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002-2014.
• Wikipedia, Primes in arithmetic progression.
• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

Cf. A000040, A001223, A014689, A014692.

Equals A359408 \ A360735.

Cf. A123556, A342309.

First column of A086800, and of A379011, last diagonal of A090321, and of A162621.

See also irregular triangles A103728, A319148, A369497.

Programs

GAP

Filtered([1..10^2],IsPrime)-2; # Muniru A Asiru, Jan 31 2018

Haskell

a040976 n = a000040 n - 2
a040976_list = map (subtract 2) a000040_list
-- Reinhard Zumkeller, Feb 22 2012

Magma

[NthPrime(n)-2: n in [1..60]]; // Vincenzo Librandi, Jan 31 2018

Maple

seq(ithprime(n)-2,n=1..100); # Muniru A Asiru, Jan 31 2018

Mathematica

Prime[Range[22]]-2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

PARI

a(n)=prime(n)-2 \\ Charles R Greathouse IV, Nov 20 2012

Formula

a(n) = A000040(n) - 2 = Sum_{i=1..n-1} A001223(i).

For n > 2: A092953(a(n)) = 1. - Reinhard Zumkeller, Nov 10 2012

If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(16) = 2 and 16 is not a term. - Bernard Schott, Feb 19 2023

a(n) = Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(A000720(k)/n)). [Ruiz and Sondow]. - Elias Alejandro Angulo Klein, Apr 09 2024