A119690 -id:A119690 - cp's OEIS frontend

A175567 (n!)^2 modulo n(n+1)/2.

0, 1, 0, 6, 0, 15, 0, 0, 0, 45, 0, 66, 0, 0, 0, 120, 0, 153, 0, 0, 0, 231, 0, 0, 0, 0, 0, 378, 0, 435, 0, 0, 0, 0, 0, 630, 0, 0, 0, 780, 0, 861, 0, 0, 0, 1035, 0, 0, 0, 0, 0, 1326, 0, 0, 0, 0, 0, 1653, 0, 1770, 0, 0, 0, 0, 0, 2145, 0, 0, 0, 2415, 0, 2556, 0, 0, 0, 0, 0, 3003, 0, 0, 0, 3321

Offset: 1

John W. Layman, Jul 12 2010

nonn

It appears that if n is one less than an odd prime then (n!)^2 modulo n(n+1)/2 is n(n-1)/2 else 0. This result appears to hold for any even power of n!. See A119690 for similar results related to odd powers of n!.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A119690, A169690, A175553.

Table[Mod[(n!)^2, (n^2 + n)/2], {n, 100}] (* Vincenzo Librandi, Jul 10 2014 *)
Table[PowerMod[n!,2,(n(n+1))/2],{n,100}] (* Harvey P. Dale, Aug 27 2016 *)

a(n) = (n!)^2 % (n*(n+1)/2); \\ Michel Marcus, Jul 09 2014

A226718 n! mod tetrahedral(n), that is A000142(n) mod A000292(n).

Original entry on oeis.org

0, 2, 6, 4, 15, 48, 0, 0, 45, 120, 66, 168, 0, 0, 120, 288, 153, 360, 0, 0, 231, 528, 0, 0, 0, 0, 378, 840, 435, 960, 0, 0, 0, 0, 630, 1368, 0, 0, 780, 1680, 861, 1848, 0, 0, 1035, 2208, 0, 0, 0, 0, 1326, 2808, 0, 0, 0, 0, 1653, 3480, 1770, 3720, 0, 0, 0, 0, 2145, 4488

Offset: 1

Alex Ratushnyak, Jun 15 2013

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000142, A000292, A119690.

A226718 := proc(n)
    n! mod ( n*(n+1)*(n+2)/6) ;
end proc: # R. J. Mathar, Jun 18 2013

Table[Mod[n!, n (n + 1) (n + 2)/6], {n, 66}] (* Ivan Neretin, May 18 2015 *)

f = 1
for i in range(1, 100):
    f *= i
    print(f % (i*(i+1)*(i+2)//6), end=', ')

a(n) = n! mod (n*(n+1)*(n+2)/6).

For n>4: if neither n+1 nor n+2 is prime, then a(n)=0. Otherwise, a(n)=n(n+1)/2 for odd n and a(n)=n(n+2) for even n. - Ivan Neretin, May 18 2015

A120387 c(n) mod b(n) where c(n) = (n-1)! and b(n) = Sum_{i=1..n} i.

Original entry on oeis.org

0, 1, 2, 6, 9, 15, 20, 0, 0, 45, 54, 66, 77, 0, 0, 120, 135, 153, 170, 0, 0, 231, 252, 0, 0, 0, 0, 378, 405, 435, 464, 0, 0, 0, 0, 630, 665, 0, 0, 780, 819, 861, 902, 0, 0, 1035, 1080, 0, 0, 0, 0, 1326, 1377, 0, 0, 0, 0, 1653, 1710, 1770, 1829, 0, 0, 0, 0, 2145, 2210, 0, 0, 2415

Offset: 1

Paolo P. Lava, Jun 30 2006

			a(5) = (5-1)! mod (1+2+3+4+5) = 24 mod 15 = 9.

Cf. A000217, A116536.

Cf. A226718, A119690.

P:=proc(n) local i,k; for i from 1 by 1 to n do print((i-1)! mod sum('k','k'=0..i)); od; end: P(100);

For n>1: if neither n nor n+1 is prime, then a(n)=0. Otherwise, a(n)=n(n-1)/2 - 1 for odd n and a(n)=n(n-1)/2 for even n. - Ivan Neretin, May 29 2015

A166260 a(n) = A089026(n) - 1.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 6, 0, 0, 0, 10, 0, 12, 0, 0, 0, 16, 0, 18, 0, 0, 0, 22, 0, 0, 0, 0, 0, 28, 0, 30, 0, 0, 0, 0, 0, 36, 0, 0, 0, 40, 0, 42, 0, 0, 0, 46, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 58, 0, 60, 0, 0, 0, 0, 0, 66, 0, 0, 0, 70, 0, 72, 0, 0, 0, 0, 0, 78, 0, 0, 0, 82, 0, 0, 0, 0, 0, 88, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Oct 10 2009

nonn

Same as A061006 except for a(4) = 0 (Wilson's Theorem). - Georg Fischer, Oct 12 2018

Cf. A010051, A061006, A089026.

Cf. A119690. - R. J. Mathar, Oct 16 2009

a(n) = if (isprime(n), n-1, 0); \\ Michel Marcus, Oct 12 2018

a(n) = (n-1) * A010051(n). - Wesley Ivan Hurt, Oct 12 2018

A175624 a(n) = n! modulo n(n+1)(n+2)/3.

Original entry on oeis.org

1, 2, 6, 24, 50, 48, 0, 0, 210, 120, 352, 168, 0, 0, 800, 288, 1122, 360, 0, 0, 2002, 528, 0, 0, 0, 0, 4032, 840, 4930, 960, 0, 0, 0, 0, 8400, 1368, 0, 0, 11440, 1680, 13202, 1848, 0, 0, 17250, 2208, 0, 0, 0, 0, 24752, 2808, 0, 0, 0, 0, 34162, 3480, 37760, 3720, 0, 0, 0, 0

Offset: 1

John W. Layman, Jul 27 2010

nonn

It appears that a(1)=1, a(2)=2, a(3)=6, and, for n>3, a(n) = n*(n+2) if n+1 is prime, else a(n) = n*(n+1)*(n+5)/6 if n+2 is prime, else a(n)=0. This has been verified for n up to 1000.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A061006, A072230, A119690, A120387, A175567.

[Factorial(n) mod (2*Binomial(n+2,3)): n in [1..80]]; // G. C. Greubel, Apr 12 2024

Table[Mod[(n!), (n^3 + 3 n^2 + 2 n)/3], {n, 100}] (* Vincenzo Librandi, Jul 10 2014 *)

a(n) = n! % (n*(n+1)*(n+2)/3); \\ Michel Marcus, Jul 09 2014

[factorial(n)%(2*binomial(n+2,3)) for n in range(1,81)] # G. C. Greubel, Apr 12 2024

cp's OEIS Frontend

A175567 (n!)^2 modulo n(n+1)/2.

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A226718 n! mod tetrahedral(n), that is A000142(n) mod A000292(n).

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A120387 c(n) mod b(n) where c(n) = (n-1)! and b(n) = Sum_{i=1..n} i.

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A166260 a(n) = A089026(n) - 1.

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A175624 a(n) = n! modulo n*(n+1)*(n+2)/3.

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A175624 a(n) = n! modulo n(n+1)(n+2)/3.