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Swinging Wilson quotients ( A163210) which are primes.
+20
3
3, 23, 71, 757, 30671, 1383331, 245273927, 3362110459, 107752663194272623, 5117886516250502670227, 34633371587745726679416744736000996167729085703, 114326045625240879227044995173712991937709388241980425799
EXAMPLE
The quotient (252+1)/11 = 23 is a swinging Wilson quotient and a prime, so 23 is a member.
MATHEMATICA
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 *)
Wilson quotients ( A007619) which are primes.
+10
5
5, 103, 329891, 10513391193507374500051862069
LINKS
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.
EXAMPLE
The quotient (720+1)/7 = 103 is a Wilson quotient and a prime, so 103 is a member.
MAPLE
A163212 := n -> select(isprime, WQ(factorial, p->1, n)):
PROG
(PARI) forprime(p=2, 1e4, a=((p-1)!+1)/p; if(ispseudoprime(a), print1(a, ", "))) \\ Felix Fröhlich, Aug 03 2014
This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.
+10
5
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
COMMENTS
This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.
EXAMPLE
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;
...
MATHEMATICA
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 *)
PROG
(Sage)
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)
Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function ( A056040).
+10
4
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
COMMENTS
If this is zero, p is a swinging Wilson prime.
EXAMPLE
The swinging Wilson quotient related to the 5th prime is (252+1)/11=23, so the 5th term is 23 mod 11 = 1.
MAPLE
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);
MATHEMATICA
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 *)
n*a(n) provides the Moebius transform of signed central binomial coefficients.
+10
3
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
COMMENTS
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
EXAMPLE
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 + ...
MAPLE
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:
MATHEMATICA
a[ n_] := If[ n < 1, 0, DivisorSum[ n, MoebiusMu[ n/#] (-1)^Quotient[ #, 2] Binomial[ # - 1, Quotient[ # - 1, 2]] &] / n]; (* Michael Somos, Sep 14 2015 *)
PROG
(Sage)
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)]
(PARI) {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 */
a(n) = binomial(2*n-2, n-1) + (-1)^n.
+10
3
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
COMMENTS
An odd prime p divides a((p+1)/2) which gives A163210.
FORMULA
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
MAPLE
a := n -> binomial(2*n-2, n-1) + (-1)^n: seq(a(n), n=1..27);
MATHEMATICA
a[n_] := Binomial[2n-2, n-1] + (-1)^n; a[Range[1, 27]]
PROG
(Magma) [Binomial(2*n-2, n-1) + (-1)^n: n in [1..30]]; // G. C. Greubel, Jul 14 2024 (SageMath)
def A286033(n): return binomial(2*n-2, n-1) + (-1)^n (Maxima) 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 */
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