A352014
a(n) = Sum_{p|n, p prime} (-1)^(n/p+1) * (n-1)!/(p-1)!.
Original entry on oeis.org
0, 1, 1, -6, 1, 60, 1, -5040, 20160, 347760, 1, -59875200, 1, 6218372160, 47221574400, -1307674368000, 1, 177843714048000, 1, -126713646259200000, 1219830034655232000, 51090928092415411200, 1, -38778025108327464960000, 25852016738884976640000
Offset: 1
-
a[n_] := Sum[(-1)^(n/p + 1)*(n - 1)!/(p - 1)!, {p, FactorInteger[n][[;; , 1]]}]; a[1] = 0; Array[a, 25] (* Amiram Eldar, Oct 04 2023 *)
-
a(n) = sumdiv(n, d, isprime(d)*(-1)^(n/d+1)*(n-1)!/(d-1)!);
-
my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*log(1+x^k)/k!))))
A352004
Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k))^(1/prime(k)!).
Original entry on oeis.org
1, 0, 1, 1, 9, 11, 295, 337, 13041, 45550, 1043211, 3359786, 150500053, 440947300, 23238057921, 145733451241, 5097210717873, 29028404123105, 1710073810205317, 8663532297784519, 574604164708374861, 5108822296820280256, 246335435270285805885
Offset: 0
-
my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^(isprime(k)/k!))))
-
my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, isprime(d)*(k-1)!/(d-1)!)*x^k/k!))))
A352058
a(n) = Sum_{p|n, p prime} n!/(p!*(n/p)!).
Original entry on oeis.org
0, 1, 1, 6, 1, 120, 1, 840, 10080, 30240, 1, 3659040, 1, 17297280, 3632428800, 259459200, 1, 1490852563200, 1, 845092863014400, 3379030566912000, 28158588057600, 1, 2565331847811532800, 1077167364120207360000, 64764752532480000, 5001134190558105600000
Offset: 1
-
a[1] = 0; a[n_] := Plus @@ (n!/((p=FactorInteger[n][[;;,1]])!*(n/p)!)); Array[a, 30] (* Amiram Eldar, Mar 02 2022 *)
-
a(n) = my(f=factor(n)); sum(k=1, #f~, n!/(f[k, 1]!*(n/f[k, 1])!));
-
my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*(exp(x^k)-1)/k!))))
A352059
a(n) = Sum_{p|n, p prime} (n-1)!/(n/p-1)!.
Original entry on oeis.org
0, 1, 2, 6, 24, 180, 720, 840, 20160, 378000, 3628800, 6985440, 479001600, 6235669440, 47221574400, 259459200, 20922789888000, 2972883513600, 6402373705728000, 20274518622758400, 1219830034655232000, 51090956251003468800, 1124000727777607680000
Offset: 1
-
a[1] = 0; a[n_] := Plus @@ ((n-1)!/(n/FactorInteger[n][[;;,1]] - 1)!); Array[a, 25] (* Amiram Eldar, Mar 02 2022 *)
-
a(n) = my(f=factor(n)); sum(k=1, #f~, (n-1)!/(n/f[k, 1]-1)!);
-
my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*(exp(x^k)-1)/k))))
A352060
a(n) = (n - 1)! * omega(n), where omega(n) = number of distinct primes dividing n (A001221).
Original entry on oeis.org
0, 1, 2, 6, 24, 240, 720, 5040, 40320, 725760, 3628800, 79833600, 479001600, 12454041600, 174356582400, 1307674368000, 20922789888000, 711374856192000, 6402373705728000, 243290200817664000, 4865804016353280000, 102181884343418880000, 1124000727777607680000
Offset: 1
-
a[n_] := (n-1)! * PrimeNu[n]; Array[a, 25] (* Amiram Eldar, Mar 02 2022 *)
-
a(n) = (n-1)!*omega(n);
-
my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(-sum(k=1, N, isprime(k)*log(1-x^k)/k))))
Showing 1-5 of 5 results.