A104636 -id:A104636 - cp's OEIS frontend

A104635 Odd n such that 2*n+1 is prime.

1, 3, 5, 9, 11, 15, 21, 23, 29, 33, 35, 39, 41, 51, 53, 63, 65, 69, 75, 81, 83, 89, 95, 99, 105, 111, 113, 119, 125, 131, 135, 141, 153, 155, 165, 173, 179, 183, 189, 191, 209, 215, 219, 221, 231, 233, 239, 243, 245, 249, 251, 261, 273, 281, 285, 293, 299

Offset: 1

Zak Seidov, Mar 18 2005

Also: Numbers k such that 2k+1 is in A002145, i.e., a Gaussian prime. - M. F. Hasler, Feb 25 2011

Also: Number of quadratic residues modulo A002145(n). - M. F. Hasler, Feb 25 2011

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002145, A005097, A104636.

[n: n in [1..500 by 2] | IsPrime(2*n+1)]; // Vincenzo Librandi, Aug 14 2018

Select[Range[1,301,2],PrimeQ[2#+1]&] (* Harvey P. Dale, May 08 2012 *)

forstep( k=1,250,2, isprime(2*k+1) && print1(k", ")) \\ M. F. Hasler, Feb 25 2011

forprime( p=1,500, p%4==3 || next; print1(p\2", ")) \\ M. F. Hasler, Feb 25 2011

a(n) = floor(A002145(n)/2). - M. F. Hasler, Feb 25 2011

A282706 Smallest prime factor of A020549(n) = (n!)^2 + 1.

Original entry on oeis.org

2, 2, 5, 37, 577, 14401, 13, 101, 17, 131681894401, 13168189440001, 1593350922240001, 101, 38775788043632640001, 29, 1344169, 149, 9049, 37, 710341, 41, 61, 337, 509, 384956219213331276939737002152967117209600000001, 941

Offset: 0

N. J. A. Sloane, Feb 26 2017

nonn

By construction, for n >= 2, a(n) == 1 (mod 4) and a(n) > n.
From Robert Israel, Mar 08 2017: (Start)
a(n) = A020549(n) for n in A046029.
a(n) <= 2*n+1 if n is in A104636.
The first member of A104636 for which a(n) < 2*n+1 is 48.
a(a(n)-n-1) = a(n). (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.

Hugo Pfoertner, Table of n, a(n) for n = 0..74
Apoloniusz Tyszka, On ZFC-formulae phi(x) for which we know a non-negative integer n such that max({x, element of N, phi(x)}) <= n if the set {x, element of N, phi(x)} is finite, 2019.
Apoloniusz Tyszka, Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
Andrew Walker, Table of factors of (n!)^2+1.

Cf. A020549, A046029, A083340, A083341, A104636, A282705.

[2] cat [Min(PrimeFactors(Factorial(n)^2 + 1)):n in[1..25]]; // Vincenzo Librandi, Feb 28 2017

f:= proc(n) local a;
  a:= min(map(proc(t) if t[1]::integer then t[1] fi end proc, ifactors((n!)^2+1,easy)[2]));
if a = infinity then
   a:= traperror(timelimit(60, min(map(t -> t[1], ifactors((n!)^2+1)[2]))));
fi;
  a
end proc:
map(f, [$0..36]); # Robert Israel, Mar 08 2017

Join[{2}, Array[FactorInteger[(#!)^2 + 1][[1, 1]]&, {25}]] (* Vincenzo Librandi, Feb 28 2017 *)

More terms from Vincenzo Librandi, Feb 28 2017

A211173 (2n)!^n (modulo 2n+1).

Original entry on oeis.org

0, 2, 1, 6, 0, 10, 1, 0, 1, 18, 0, 22, 0, 0, 1, 30, 0, 0, 1, 0, 1, 42, 0, 46, 0, 0, 1, 0, 0, 58, 1, 0, 0, 66, 0, 70, 1, 0, 0, 78, 0, 82, 0, 0, 1, 0, 0, 0, 1, 0, 1, 102, 0, 106, 1, 0, 1, 0, 0, 0, 0, 0, 0, 126, 0, 130, 0, 0, 1, 138, 0

Offset: 0

Larry Riddle (LRiddle(AT)AgnesScott.edu) and Robert G. Wilson v, Jan 31 2013

a(n)= 0, 1 or 2n. In fact, a(n) = 0 iff n belongs to A047845, a(n) = 1 iff n belongs to A104636 and a(n) = 2n iff n belongs to A104635.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept. Jul, 1999.

Cf. A047845, A104635, A104636.

f[n_] := Mod[((2 n)!)^n, 2 n + 1]; Array[f, 70]
Table[PowerMod[(2n)!,n,2n+1],{n,0,70}] (* Harvey P. Dale, Nov 02 2019 *)

a(n)=if(isprime(2*n+1),if(n%2,2*n,1),0) \\ Charles R Greathouse IV, Feb 07 2013

a(n) = (2n)!^n (modulo 2n+1).

A237053 Smallest number k such that some subset of n+1..n+k can be summed and added to n to produce a prime.

Original entry on oeis.org

2, 1, 0, 0, 3, 0, 1, 0, 1, 1, 3, 0, 3, 0, 1, 1, 3, 0, 1, 0, 1, 1, 3, 0, 4, 3, 1, 5, 3, 0, 1, 0, 3, 1, 3, 1, 1, 0, 3, 1, 3, 0, 3, 0, 1, 3, 4, 0, 1, 3, 1, 1, 3, 0, 1, 3, 1, 5, 3, 0, 5, 0, 3, 1, 3, 1, 4, 0, 1, 1, 6, 0, 4, 0, 1, 1, 3, 3, 1, 0, 3, 1, 3, 0, 3, 3, 1, 5, 3, 0, 1, 3, 3, 3, 3, 1, 1

Offset: 0

Franklin T. Adams-Watters, Feb 03 2014

a(n) = 0 iff n is prime.
a(n) = 2 only for n=0; the only possible sums for k=2 are n+(n+2) = 2n+2, divisible by 2, and n+(n+1)+(n+2) = 3n+3, divisible by 3.
There are infinitely many 1's in the sequence; if p > 5 is a prime == 1 (mod 4), a((p-1)/2) = 1.
Conjecture: every nonnegative integer except 2 occurs infinitely often in the sequence.

			If n is prime, sum({n}) is prime, so we can take k = 0, whence n+1..n+0 is empty, so a(n) = 0.
6 is not prime, but 6+7 = 13 is prime, so a(6) = 1.
4 is not prime, and 4+5 is not prime, but 4+7 = 11 and 4+6+7 = 17 are prime; either of these suffices to make a(4) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..100000

Cf. A000040, A005097, A104636, A077654, A089306.

b:= (n, i, t)-> isprime(n) or t>0 and
    (b(n, i+1, t-1) or b(n+i, i+1, t-1)):
a:= proc(n) local k;
      for k from 0 while not b(n, n+1, k) do od; k
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014

b[n_, i_, t_] := PrimeQ[n] || t > 0 && (b[n, i+1, t-1] || b[n+i, i+1, t-1]);
a[n_] := Module[{k}, For[k = 0, !b[n, n+1, k], k++]; k];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 04 2025, after Alois P. Heinz *)

A336257 a(n) = Catalan(n) mod (2*n+1).

Original entry on oeis.org

0, 1, 2, 5, 5, 9, 2, 9, 2, 17, 17, 21, 12, 22, 2, 29, 18, 30, 2, 30, 2, 41, 30, 45, 9, 21, 2, 54, 53, 57, 2, 28, 38, 65, 42, 69, 2, 64, 70, 77, 5, 81, 80, 33, 2, 14, 27, 45, 2, 36, 2, 101, 87, 105, 2, 78, 2, 34, 75, 6, 101, 45, 62, 125, 39, 129, 74, 60, 2, 137, 90

Offset: 0

Michel Marcus, Jul 15 2020

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Arturo Merino, Ondrej Micka, and Torsten Mütze, On a combinatorial generation problem of Knuth, arXiv:2007.07164 [math.CO], 2020. See p. 43.

Cf. A000108, A059288, A067770, A246714.

Cf. A104635, A104636.

a:= n-> binomial(2*n, n)/(n+1) mod (2*n+1):
seq(a(n), n=0..80);  # Alois P. Heinz, Jul 16 2020

C(n)=binomial(2*n, n)/(n+1);
a(n) = C(n) % (2*n+1);

A336257_list, c = [0,1], 1
for n in range(2,10001):
    c = c*(4*n-2)//(n+1)
    A336257_list.append(c % (2*n+1)) # Chai Wah Wu, Jul 16 2020

a(n) = 2 for n in A104636.

a(n) = 2*n-1 for n in A104635.

cp's OEIS Frontend

A104635 Odd n such that 2*n+1 is prime.

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A282706 Smallest prime factor of A020549(n) = (n!)^2 + 1.

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A211173 (2n)!^n (modulo 2n+1).

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A237053 Smallest number k such that some subset of n+1..n+k can be summed and added to n to produce a prime.

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A336257 a(n) = Catalan(n) mod (2*n+1).

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