A056040 -id:A056040 - cp's OEIS frontend

A196748 Numbers n such that 3 and 5 do not divide swing(n) = A056040(n).

0, 1, 2, 20, 24, 54, 60, 61, 62, 72, 73, 74, 504, 510, 511, 512, 560, 564, 1512, 1513, 1514, 1520, 1620, 1621, 1622, 6320, 6324, 6372, 6373, 6374, 6500, 6504, 6552, 6553, 6554, 6560, 13122, 13123, 13124, 13770, 13771, 13772, 13824, 13850, 15072, 15073, 15074

Offset: 1

Peter Luschny, Oct 06 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
P. Erdos, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196747, A196749, A196750.

# The function Search is defined in A196747.
A196748_list := n -> Search(n,[3,5]):  # n is a search limit

(* A naive solution *) sf[n_] := n!/Quotient[n, 2]!^2; Select[Range[0, 16000], !Divisible[sf[#], 3] && !Divisible[sf[#], 5] &] (* Jean-François Alcover, Jun 28 2013 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=valp(n,3)==2*valp(n\2,3) && valp(n,5)==2*valp(n\2,5) \\ Charles R Greathouse IV, Feb 02 2016

A352363 Triangle read by rows. The incomplete Bell transform of the swinging factorials A056040.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 6, 50, 35, 10, 1, 0, 30, 166, 225, 85, 15, 1, 0, 20, 756, 1246, 735, 175, 21, 1, 0, 140, 2932, 7588, 5761, 1960, 322, 28, 1, 0, 70, 11556, 45296, 46116, 20181, 4536, 546, 36, 1

Offset: 0

Peter Luschny, Mar 15 2022

			Triangle starts:
[0] 1;
[1] 0,   1;
[2] 0,   1,     1;
[3] 0,   2,     3,     1;
[4] 0,   6,    11,     6,     1;
[5] 0,   6,    50,    35,    10,     1;
[6] 0,  30,   166,   225,    85,    15,    1;
[7] 0,  20,   756,  1246,   735,   175,   21,   1;
[8] 0, 140,  2932,  7588,  5761,  1960,  322,  28, 1;
[9] 0,  70, 11556, 45296, 46116, 20181, 4536, 546, 36, 1;

Cf. A056040, A352364 (row sums), A352365 (alternating row sums).

Cf. A352366, A352369.

SwingNumber := n -> n! / iquo(n, 2)!^2:
for n from 0 to 9 do
seq(IncompleteBellB(n, k, seq(SwingNumber(j), j = 0..n)), k = 0..n) od;

Given a sequence s let s|n denote the initial segment s(0), s(1), ..., s(n).

(T(s))(n, k) = IncompleteBellPolynomial(n, k, s|n), where s(n) = n!/floor(n/2)!^2.

A163077 Numbers k such that k$ + 1 is prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 14, 15, 24, 27, 31, 38, 44, 45, 49, 67, 76, 92, 99, 119, 124, 133, 136, 139, 144, 168, 171, 185, 265, 291, 332, 368, 428, 501, 631, 680, 689, 696, 765, 789, 890, 1034, 1233, 1384, 1517, 1615, 1634, 1809, 2632, 2762, 3925, 4419, 5108, 5426

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			0$ + 1 = 1 + 1 = 2 is prime, so 0 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A163078, A163079, A163080.

a := proc(n) select(x -> isprime(A056040(x)+1),[$0..n]) end:

fQ[n_] := PrimeQ[1 + 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]]; Select[ Range[0, 8660], fQ] (* Robert G. Wilson v, Aug 09 2010 *)

is(k) = ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(45)-a(56) from Robert G. Wilson v, Aug 09 2010

A163078 Numbers k such that k$ - 1 is prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 13, 15, 18, 30, 35, 39, 41, 47, 49, 58, 83, 86, 102, 111, 137, 151, 195, 205, 226, 229, 317, 319, 321, 368, 389, 426, 444, 477, 534, 558, 567, 738, 804, 882, 1063, 1173, 1199, 1206, 1315, 1624, 1678, 1804, 2371, 2507, 2541, 2844, 3084, 3291

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			4$ - 1 = 6 - 1 = 5 is prime, so 4 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002982, A056040, A163077, A163079, A163080.

a := proc(n) select(x -> isprime(A056040(x)-1),[$0..n]) end:

fQ[n_] := PrimeQ[ -1 + 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]]; Select[ Range@ 3647, fQ] (* Robert G. Wilson v, Aug 09 2010 *)

is(k) = ispseudoprime(k!/(k\2)!^2-1); \\ Jinyuan Wang, Mar 22 2020

a(42)-a(54) from Robert G. Wilson v, Aug 09 2010

A163080 Primes p such that p$ - 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

3, 5, 7, 13, 41, 47, 83, 137, 151, 229, 317, 389, 1063, 2371, 6101, 7873, 13007, 19603

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163078.

			3 is prime and 3$ - 1 = 5 is prime, so 3 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A056040, A103317, A163079, A163078.

a := proc(n) select(isprime,select(k -> isprime(A056040(k)-1),[$0..n])) end:

sf[n_] := n!/Quotient[n, 2]!^2; Select[Prime /@ Range[200], PrimeQ[sf[#] - 1] &] (* Jean-François Alcover, Jun 28 2013 *)

is(k) = isprime(k) && ispseudoprime(k!/(k\2)!^2-1); \\ Jinyuan Wang, Mar 22 2020

a(14)-a(18) from Jinyuan Wang, Mar 22 2020

A163083 Primes of the form k$ + 1 which are the greater of twin primes. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

7, 31, 51481, 1580132580471901

Offset: 1

Peter Luschny, Jul 21 2009

nonn

Subsequence of A163075 and of A006512.

Jinyuan Wang, Table of n, a(n) for n = 1..5
Peter Luschny, Swinging Primes.

Cf. A056040, A006512, A163075.

a := proc(n) select(s->isprime(s) and isprime(s-2), map(k -> A056040(k)+1,[$4..n])) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; Do[ If[ PrimeQ[p = sf[n] + 1] && PrimeQ[p - 2], Print["n = ", n, " p = ", p]], {n, 1, 400}] (* Jean-François Alcover, Jul 29 2013 *)

A163641 The radical of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 10, 70, 70, 210, 42, 462, 462, 6006, 858, 4290, 4290, 72930, 24310, 461890, 92378, 1939938, 176358, 4056234, 1352078, 6760390, 520030, 1560090, 222870, 6463230, 6463230, 200360130

Offset: 0

Peter Luschny, Aug 02 2009

nonn

The radical of n$ is the product of the prime numbers dividing n$. It is the largest squarefree divisor of n$, and so also described as the squarefree kernel of n$.

			11$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 2*3*7*11 = 462.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Bisections give: A080397 (even part), A163640 (odd part).

Cf. A056040.

a := proc(n) local p; mul(p,p=numtheory[factorset](n!/iquo(n,2)!^2)) end:

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *)

a(n) = rad(n$).

A163865 The binomial transform of the swinging factorial (A056040).

Original entry on oeis.org

1, 2, 5, 16, 47, 146, 447, 1380, 4251, 13102, 40343, 124136, 381625, 1172198, 3597401, 11031012, 33798339, 103477590, 316581567, 967900224, 2957316429, 9030317478, 27558851565, 84059345244, 256265811333, 780885245826, 2378410969977, 7241027262280

Offset: 0

Peter Luschny, Aug 06 2009

nonn

a(n) = Sum_{k=0..n} binomial(n,k) * k$, where k$ denotes the swinging factorial of k (A056040). The swinging analog to the number of arrangements, the binomial transform of the factorial (A000522).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A000522, A163650.

a := proc(n) local k: add(binomial(n,k)*(k!/iquo(k, 2)!^2),k=0..n) end:
seq(coeff(series((1-z-4*z^2)/((1+z)*(1-3*z))^(3/2),z,28),z,n),n=0..27); # Peter Luschny, Oct 31 2013

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Sum[Binomial[n, k]*sf[k], {k, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 26 2013 *)
sf[n_] := n!/Quotient[n, 2]!^2; t[n_] := Sum[Binomial[n, k]*sf[k], {k, 0, n}]; Table[t[n], {n,0,50}] (* G. C. Greubel, Aug 06 2017 *)

x='x+O('x^50); Vec((1-x-4*x^2)/((1+x)*(1-3*x))^(3/2)) \\ G. C. Greubel, Aug 06 2017

E.g.f.: exp(x)*BesselI(0,2*x)*(1+x). - Peter Luschny, Aug 26 2012
O.g.f.: (1-x-4*x^2)/((1+x)*(1-3*x))^(3/2). - Peter Luschny, Oct 31 2013
a(n) ~ 3^(n - 1/2) * sqrt(n) / (2*sqrt(Pi)). - Vaclav Kotesovec, Nov 27 2017

A196749 Numbers n such that 3, 5 and 7 do not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 20, 1512, 1513, 1514, 6320, 6372, 6373, 6374, 6500, 15120, 15121, 15122, 15302, 40014, 119096754, 119096802, 91547225622, 91550794374

Offset: 1

Peter Luschny, Oct 06 2011

P. Erdos, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196747, A196748, A196750.

# The function Search is defined in A196747.
A196749_list := n -> Search(n,[3,5,7]):  # n is a search limit

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=valp(n,3)==2*valp(n\2,3) && valp(n,5)==2*valp(n\2,5) && valp(n,7)==2*valp(n\2,7) \\ Charles R Greathouse IV, Feb 02 2016

A196750 Numbers n such that 3, 5, 7 and 11 do not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 6320

Offset: 1

Peter Luschny, Oct 06 2011

It is conjectured that there are no other terms.

Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196747, A196748, A196749.

# The function Search is defined in A196747.
A196750_list := n -> Search(n,[3,5,7,11]):  # n is a search limit

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=valp(n,3)==2*valp(n\2,3) && valp(n,5)==2*valp(n\2,5) && valp(n,7)==2*valp(n\2,7) && valp(n,11)==2*valp(n\2,11) \\ Charles R Greathouse IV, Feb 02 2016

cp's OEIS Frontend

A196748 Numbers n such that 3 and 5 do not divide swing(n) = A056040(n).

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A352363 Triangle read by rows. The incomplete Bell transform of the swinging factorials A056040.

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A163077 Numbers k such that k$ + 1 is prime. Here '$' denotes the swinging factorial function (A056040).

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A163078 Numbers k such that k$ - 1 is prime. Here '$' denotes the swinging factorial function (A056040).

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A163080 Primes p such that p$ - 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

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A163083 Primes of the form k$ + 1 which are the greater of twin primes. Here '$' denotes the swinging factorial function (A056040).

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A163641 The radical of the swinging factorial A056040.

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A163865 The binomial transform of the swinging factorial (A056040).