A307124 a(n) is twice the square of the product of the first n primes each decreased by one.

2, 8, 128, 4608, 460800, 66355200, 16986931200, 5503765708800, 2663822603059200, 2088436920798412800, 1879593228718571520000, 2435952824419268689920000, 3897524519070829903872000000, 6875233251640943950430208000000

Offset

1,1

Comments

a(n) is also the number of pairs (c, d) for which d !== c (mod p) and d !== -c (mod p) for every prime p up to and including the n-th prime, where 1 <= c, d <= the product of the first n primes.

In some cases, c - d and c + d are prime with (c - d) + (c + d) = 2c, as in Goldbach's conjecture.

Links

Table of n, a(n) for n=1..14.

Sally M. Moite, Partition of the Primorial Square By Remainder Agreement Counts, viXra:1906.0069 (2019). Additional comments. Unedited.

Formula

From Jon E. Schoenfield, Mar 26 2019: (Start)

a(n) = 2*Product_{j=1..n} (prime(j)-1)^2.

a(n) = 2*A005867(n)^2. (End)

Example

a(4) = 2 * 1^2 * 2^2 * 4^2 * 6^2 = 4608.
For n = 2, 2 * 3 = 6, the pairs (c, d) are (1, 6), (2, 3), (3, 2), (3, 4), (4, 3), (5, 6), (6, 1) and (6, 5), so a(2) = 8.

Mathematica

a[n_]:=2 Product[(Prime[i] - 1)^2, {i, 1, n}]; Array[a, 10]
2*FoldList[Times, (Prime[Range[15]]-1)^2] (* Harvey P. Dale, Jul 20 2022 *)

Prog

(PARI) a(n) = 2*prod(k=1, n, prime(k)-1)^2; \\ Michel Marcus, Mar 27 2019
(Magma)  [2*(&*[(NthPrime(i)-1)^2: i in [1..n]]): n in [1..14]]; // Marius A. Burtea, Jun 19 2019

Crossrefs

Cf. A002110, A005867, A061742.

Sequence in context: A152922 A114977 A011822 * A111179 A178173 A058891

Adjacent sequences: A307121 A307122 A307123 * A307125 A307126 A307127

Keyword

nonn

Author

Sally Myers Moite, Mar 26 2019

Extensions

First Mathematica program edited by Harvey P. Dale, Jul 20 2022

Status

approved