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A323739 a(n) is the number of residues modulo (4*primorial(n)) of the squares of primes greater than or equal to prime(n+1). 0
2, 1, 1, 2, 6, 30, 180, 1440, 12960, 142560, 1995840 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Here, "primorial(n)" is A002110(n) = Product_{k=1..n} prime(k).

LINKS

Table of n, a(n) for n=0..10.

FORMULA

Conjecture: a(n) = 2^(1-n)*Product_{j=1..n} (prime(j)-1) for n >= 0, so a(n) = a(n-1)*(prime(n)-1)/2 for n >= 1.

From Charlie Neder, Feb 28 2019: (Start)

Conjecture is true. Since there exists a prime congruent to r modulo 4*primorial(n) for any r coprime to primorial(n), this set is precisely the set of coprime quadratic residues of 4*primorial(n). If n >= 1, each residue can be broken down into congruences modulo 8 and the first n-1 odd primes, each odd prime p has (p-1)/2 residue classes, and every combination eventually occurs, giving the formula. (End)

EXAMPLE

a(3) = 2 because, for every prime p >= prime(3+1) = 7, p^2 mod (4*2*3*5 = 120) is one of the 2 values {1, 49}:

   7^2 mod 120 =  49 mod 120 = 49

  11^2 mod 120 = 121 mod 120 =  1

  13^2 mod 120 = 169 mod 120 = 49

  17^2 mod 120 = 289 mod 120 = 49

  19^2 mod 120 = 361 mod 120 =  1

  23^2 mod 120 = 529 mod 120 = 49

  29^2 mod 120 = 841 mod 120 =  1

  ...

.

   q=(n+1)st        b =          residues p^2 mod b

n    prime    4*primorial(n)         for p >= q         a(n)

=  =========  ===============  =======================  ====

0      2      4         =   4           {0,1}             2

1      3      4*2       =   8            {1}              1

2      5      4*2*3     =  24            {1}              1

3      7      4*2*3*5   = 120           {1,49}            2

4     11      4*2*3*5*7 = 840  {1,121,169,289,361,529}    6

CROSSREFS

Cf. A002110, A005867, A240775.

Sequence in context: A319031 A022874 A022873 * A122160 A058316 A082386

Adjacent sequences:  A323736 A323737 A323738 * A323740 A323741 A323742

KEYWORD

nonn,more

AUTHOR

Jon E. Schoenfield, Feb 20 2019

STATUS

approved

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Last modified April 8 21:44 EDT 2020. Contains 333329 sequences. (Running on oeis4.)