A163210 -id:A163210 - cp's OEIS frontend

A163211 Swinging Wilson quotients (A163210) which are primes.

3, 23, 71, 757, 30671, 1383331, 245273927, 3362110459, 107752663194272623, 5117886516250502670227, 34633371587745726679416744736000996167729085703, 114326045625240879227044995173712991937709388241980425799

Offset: 1

Peter Luschny, Jul 24 2009

nonn

a(14)-a(18) certified prime by Primo 4.2.0. a(17) = A163210(569) = P1239, a(18) = A163210(787) = P1812. - Charles R Greathouse IV, Dec 11 2016

			The quotient (252+1)/11 = 23 is a swinging Wilson quotient and a prime, so 23 is a member.

G. C. Greubel, Table of n, a(n) for n = 1..16
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163210, A163213, A163212, A163209, A007619.

A163211 := n -> select(isprime,A163210(n));

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Select[PrimeQ][Table[a[n], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2016 *)

sf(n)=n!/(n\2)!^2
forprime(p=2,1e3, t=sf(p-1)\/p; if(isprime(t), print1(t", "))) \\ Charles R Greathouse IV, Dec 11 2016

A163212 Wilson quotients (A007619) which are primes.

Original entry on oeis.org

5, 103, 329891, 10513391193507374500051862069

Offset: 1

Peter Luschny, Jul 24 2009

a(5) = A007619(137), a(6) = A007619(216), a(7) = A007619(381).

Same as A122696 without its initial term 2. - Jonathan Sondow, May 19 2013

			The quotient (720+1)/7 = 103 is a Wilson quotient and a prime, so 103 is a member.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. A050299, A163211, A007619, A122696, A163210, A163213, A163209, A225906.

# WQ defined in A163210.
A163212 := n -> select(isprime,WQ(factorial,p->1,n)):

Select[Table[p = Prime[n]; ((p-1)!+1)/p, {n, 1, 15}], PrimeQ] (* Jean-François Alcover, Jun 28 2013 *)

forprime(p=2, 1e4, a=((p-1)!+1)/p; if(ispseudoprime(a), print1(a, ", "))) \\ Felix Fröhlich, Aug 03 2014

a(n) = A122696(n+1) = A007619(A225906(n)) = ((A050299(n+1)-1)!+1)/A050299(n+1). - Jonathan Sondow, May 19 2013

A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.

Original entry on oeis.org

1, -1, -1, 0, -1, 0, 0, 1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, -1, 1, 0, 0, -1, 2, -3, 2, -1, 0, 0, 1, -3, 4, 4, -3, 1, 0, 0, -1, 3, -6, 8, -6, 3, -1, 0, 0, 1, -3, 9, -13, -13, 9, -3, 1, 0, 0, -1, 4, -11, 19, -23, 19, -11, 4, -1, 0, 0, 1, -5, 13, -27, 39, 39, -27, 13, -5, 1, 0, 0, -1, 5, -17, 38, -61, 71, -61, 38, -17, 5, -1, 0

Offset: 0

F. Chapoton, Jun 22 2010

This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.

			a(0,0) = 1, a(1,0) = a(0,1) = -1.
Triangle begins:
   1;
  -1, -1;
   0, -1,  0;
   0,  1,  1,  0;
   0, -1,  1, -1,  0;
   0,  1, -1, -1,  1,  0;
   0, -1,  2, -3,  2, -1, 0;
   ...

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A178738, A178749, A022553, A131868, A163210.

b[m_, n_] := (-1)^Max[m, n]*Binomial[m+n, n]; A[m_, n_] := DivisorSum[ n+m+1, b[Floor[m/#], Floor[n/#]]*MoebiusMu[#]&]/(m+n+1); Table[A[m-n, n], {m, 0, 12}, {n, 0, m}] // Flatten (* Jean-François Alcover, Feb 23 2017, adapted from Python *)

def twisted_binomial(m, n):
    return (-1)**max(m, n) * binomial(m + n, n)
def coefficients_A(m, n):
    return sum(twisted_binomial(m // d, n // d) * moebius(d)
           for d in divisors(m + n + 1)) / (m + n + 1)
matrix(ZZ, 8, 8, coefficients_A)

Terms a(82) onward added by G. C. Greubel, Dec 10 2017

A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 9, 13, 12, 2, 19, 2, 5, 36, 6, 19, 43, 11, 47, 67, 39, 41, 70, 12, 17, 83, 88, 81, 25, 53, 91, 97, 106, 79, 43, 39, 7, 29, 73, 6, 79, 115

Offset: 1

Peter Luschny, Jul 24 2009

nonn

If this is zero, p is a swinging Wilson prime.

			The swinging Wilson quotient related to the 5th prime is (252+1)/11=23, so the 5th term is 23 mod 11 = 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163211, A002068, A163210.

WR := proc(f,r,n) map(p->(f(p-1)+r(p))/p mod p,select(isprime,[$1..n])) end:
A002068 := n -> WR(factorial,p->1,n);
A163213 := n -> WR(swing,p->(-1)^iquo(p+2,2),n);

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; Mod[(sf[p - 1] + (-1)^Floor[(p + 2)/2])/p, p]); Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Jun 28 2013 *)

sf(n)=n!/(n\2)!^2
apply(p->sf(p-1)\/p%p, primes(100)) \\ Charles R Greathouse IV, Dec 11 2016

A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -3, 4, 8, -13, -23, 39, 71, -121, -229, 400, 757, -1354, -2559, 4625, 8799, -16021, -30671, 56316, 108166, -200047, -385210, 716429, 1383331, -2585173, -5003791, 9391680, 18214565, -34318117, -66674463, 126044208, 245273927, -465067981

Offset: 1

F. Chapoton, Jun 09 2010

sign

This should be related to the Coxeter transformation for the Tamari lattices.

The source sequence is 1, -1, -2, 3, 6, -10, -20, 35, 70, -126, ... (A001405). Its Mobius transform is 1, -2, -3, 4, 5, -6, -21, 32, 72, -130, -253, 468, 923, ... and division of each term through n generates a(n). - R. J. Mathar, Jul 23 2012

			G.f. = x - x^2 - x^3 + x^4 + x^5 - x^6 - 3*x^7 + 4*x^8 + 8*x^9 - 13*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
F. Chapoton, Le module dendriforme sur le groupe cyclique, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2333-2350. In French.

Similar to A022553, A131868 and A178738.

Also related to A163210.

with(numtheory):
a:= n-> add(mobius(n/d)*[1$2, -1$2][1+irem(d, 4)]*
        binomial(d-1, iquo(d-1, 2)), d=divisors(n))/n:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 05 2013

a[ n_] := If[ n < 1, 0, DivisorSum[ n, MoebiusMu[ n/#] (-1)^Quotient[ #, 2] Binomial[ # - 1, Quotient[ # - 1, 2]] &] / n]; (* Michael Somos, Sep 14 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, moebius(n/d) * (-1)^(d\2) * binomial(d-1, (d-1)\2)) / n)}; /* Michael Somos, Dec 23 2014 */

def lam(n):
    return (-1)**binomial(n, 2) * binomial(n - 1, (n - 1) // 2)
def a(n):
    return sum(moebius(n // d) * lam(d) for d in divisors(n)) // n
[a(n) for n in range(1, 20)]

A286033 a(n) = binomial(2*n-2, n-1) + (-1)^n.

Original entry on oeis.org

0, 3, 5, 21, 69, 253, 923, 3433, 12869, 48621, 184755, 705433, 2704155, 10400601, 40116599, 155117521, 601080389, 2333606221, 9075135299, 35345263801, 137846528819, 538257874441, 2104098963719, 8233430727601, 32247603683099, 126410606437753, 495918532948103

Offset: 1

Peter Luschny, May 13 2017

An odd prime p divides a((p+1)/2) which gives A163210.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000984, A033999, A163210, A178904, A219539, A000032.

[Binomial(2*n-2, n-1) + (-1)^n: n in [1..30]]; // G. C. Greubel, Jul 14 2024

a := n -> binomial(2*n-2, n-1) + (-1)^n: seq(a(n), n=1..27);

a[n_] := Binomial[2n-2, n-1] + (-1)^n; a[Range[1,27]]

a(n):=-sum((-1)^k*binomial(2*n,n-k)*(fib(2*k+1)+fib(2*k-1)),k,1,n); /* Vladimir Kruchinin, Jan 18 2025 */

a(n) = binomial(2*n-2, n-1) + (-1)^n \\ David A. Corneth, May 13 2017

def A286033(n): return binomial(2*n-2, n-1) + (-1)^n
[A286033(n) for n in range(1,31)] # G. C. Greubel, Jul 14 2024

a(n) = A000984(n-1) + A033999(n). - David A. Corneth, May 13 2017
G.f.: -1 + x/sqrt(1 - 4*x) + 1/(1 + x). - Ilya Gutkovskiy, May 13 2017
D-finite with recurrence: -(n-1)*a(n) +2*(n-1)*a(n-1) +(7*n-17)*a(n-2) +2*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jan 27 2020
a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(2*n, n-k)*A000032(2*k). - Vladimir Kruchinin, Jan 18 2025

cp's OEIS Frontend

A163211 Swinging Wilson quotients (A163210) which are primes.

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A163212 Wilson quotients (A007619) which are primes.

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A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.

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A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

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A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

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A286033 a(n) = binomial(2*n-2, n-1) + (-1)^n.

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