A061006 -id:A061006 - cp's OEIS frontend

A061007 a(n) = -(n-1)! mod n.

0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Henry Bottomley, Apr 12 2001

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

In particular, this is identical to the isprime function A010051 except for a(4) = 2 instead of 0. This is equivalent to Wilson's theorem, (n-1)! == -1 (mod n) iff n is prime. If n = p*q with p, q > 1, then p, q < n-1 and (n-1)! will contain the two factors p and q, unless p = q = 2 (if p = q > 2 then also 2p < n-1, so there are indeed two factors p in (n-1)!), whence (n-1)! == 0 (mod n). - M. F. Hasler, Jul 19 2024

			a(4) = 2 since -(4 - 1)! = -6 = 2 mod 4.
a(5) = 1 since -(5 - 1)! = -24 = 1 mod 5.
a(6) = 0 since -(6 - 1)! = -120 = 0 mod 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positive for all but the first term of A046022.

Cf. A000040 (the primes), A000142, A010051 (isprime function), A055976, A061006, A061008, A061009.

Table[Mod[-(n - 1)!, n], {n, 100}] (* Alonso del Arte, Mar 20 2014 *)

A061007(n) = ((-((n-1)!))%n); \\ Antti Karttunen, Aug 27 2017

apply( {A061007(n) = !(n-1)!%n}, [0..99]) \\ M. F. Hasler, Jul 19 2024

from sympy import isprime
def A061007(n): return 2 if n == 4 else int(isprime(n)) # Chai Wah Wu, Mar 22 2023

a(4) = 2, a(p) = 1 for p prime, a(n) = 0 otherwise. Apart from n = 4, a(n) = A010051(n) = A061006(n)/(n-1).

A061008 a(n) = Sum_{j=1..n} (-(n-1)! mod n).

Original entry on oeis.org

0, 1, 2, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 23, 23, 23

Offset: 1

Henry Bottomley, Apr 12 2001

nonn

			a(6) = 5 since (-1 mod 1) + (-1 mod 2) + (-2 mod 3) + (-6 mod 4) + (-24 mod 5) + (-120 mod 6) = 0 + 1 + 1 + 2 + 1 + 0 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A000142, A061006, A061007, A061009.

[0,1,2] cat [ 2+#PrimesUpTo(n): n in [4..200] ]; // Vincenzo Librandi, Aug 11 2017

Join[{0, 1, 2}, a[n_]:= 2 + PrimePi[n]; Table[a[n], {n, 4, 100}]] (* Vincenzo Librandi, Aug 11 2017 *)

a(n) = a(n-1) + A061007(n) = A061009(n) + 2. For n > 3, a(n) = pi(n) + 2 where pi(n) = A000720(n) is the number of primes less than or equal to n.

A061009 a(n) = -2 + Sum_{j=1..n} (-(n-1)!) mod n.

Original entry on oeis.org

-2, -1, 0, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24

Offset: 1

Henry Bottomley, Apr 12 2001

sign

			a(6)=3 since -2 + (-1 mod 1) + (-1 mod 2) + (-2 mod 3) + (-6 mod 4) + (-24 mod 5) + (-120 mod 6) = -2 + 0 + 1 + 1 + 2 + 1 + 0 = 3.

Cf. A000040, A000142, A061006, A061007, A061008.

a(n) = a(n-1) + A061007(n) = A061008(n) - 2. For n > 3, a(n) = pi(n) = A000720(n) where pi(n) is the number of primes less than or equal to n.

A119688 a(n) = n!! mod (n+1).

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 1, 6, 5, 1, 3, 8, 7, 0, 1, 13, 9, 1, 15, 0, 11, 1, 9, 0, 13, 0, 7, 17, 15, 1, 1, 0, 17, 0, 27, 6, 19, 0, 25, 9, 21, 1, 11, 0, 23, 46, 33, 0, 25, 0, 39, 30, 27, 0, 49, 0, 29, 58, 15, 50, 31, 0, 1, 0, 33, 1, 51, 0, 35, 1, 9, 27, 37, 0, 19, 0, 39, 78, 65, 0, 41, 82, 63, 0

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jun 09 2006

The double factorial used here is A006882, a(n) = n*a(n-2). Bisections of A006882 are A000165 and A001147.

			5!! = 5*3*1 = 15, a(5) = 15 mod (5+1) = 3.
6!! = 6*4*2 = 48, a(6) = 48 mod (6+1) = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Double Factorial

Cf. A006882, A000165, A001147, A061006.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(n) mod (n+1): n in [1..100] ]; // Klaus Brockhaus, Feb 15 2011

P:= proc(n) local i, j, k, s; for k from 1 by 1 to n do i:=k; j:=k-2; while j >0 do i:=i*j; j:=j-2; od: s:=i mod (k+1); print(s); od: end: P(100);
## another version:
a:= proc(n) local t, m;
       if irem (n, 2)=1 or n<14 or isprime(n+1)
       then t:= 1;
            for m from n by -2 while m>1 do
              t:= (t*m) mod (n+1)
            od; t
       else 0 fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 15 2011

Table[Mod[n!!,n+1],{n,100}] (* Zak Seidov, Feb 15 2011 *)

a(n) = prod(i=0, (n-1)\2, n - 2*i) % (n+1); \\ after PARI for A006882; Michel Marcus, Aug 22 2016

a(63) corrected, a(64) inserted by Klaus Brockhaus, Feb 15 2011

A238002 Count with multiplicity of prime factors of n in (n - 1)!.

Original entry on oeis.org

0, 0, 1, 0, 4, 0, 4, 2, 8, 0, 12, 0, 11, 7, 11, 0, 21, 0, 19, 10, 19, 0, 28, 4, 23, 10, 26, 0, 44, 0, 26, 16, 32, 11, 47, 0, 35, 19, 43, 0, 61, 0, 42, 28, 42, 0, 63, 6, 56, 24, 50, 0, 72, 16, 58, 28, 54, 0, 94, 0, 57, 37, 57, 18, 98, 0, 67, 33, 91, 0, 99, 0, 71, 50, 74, 17, 113, 0, 92

Offset: 2

Alonso del Arte, Feb 16 2014

			a(4) = 1 because 3! = 6 = 2 * 3, which has one prime factor (2) in common with 4.
a(5) = 0 because gcd(4!, 5) = 1.
a(6) = 4 because 5! = 120 = 2^3 * 3 * 5, which has four factors (2 thrice and 3 once) in common with 6.

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A061006, A181569, A226198, A022559.

with(numtheory):
a:= n-> add(add(`if`(i[1] in factorset(n), i[2], 0),
        i=ifactors(j)[2]), j=1..n-1):
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 17 2014

cmpf[n_]:=Count[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[ (n-1)!]], ?( MemberQ[Transpose[FactorInteger[n]][[1]],#]&)]; Array[cmpf,80] (* _Harvey P. Dale, Jan 23 2016 *)

a(n) = {nm = (n-1)!; fn = factor(n); sum (i=1, #fn~, valuation(nm, fn[i,1]));} \\ Michel Marcus, Mar 15 2014

m=100 # change n for more terms
[sum(valuation(factorial(n-1),p) for p in prime_divisors(n) if p in prime_divisors(factorial(n-1))) for n in [2..m]] # Tom Edgar, Mar 14 2014

a(p) = 0 for p prime.

a(2n) > a(2n + 1) for all n > 2.

A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 2, 3, 2, 1, 4, 0, 0, 0, 1, 0, 4, 3, 1, 3, 1, 6, 0, 0, 0, 0, 0, 1, 0, 4, 4, 1, 0, 2, 4, 1, 0, 0, 8, 0, 3, 0, 6, 0, 1, 0, 6, 0, 0, 5, 3, 0, 0, 5, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 6, 11, 6, 3, 6, 5, 6, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 0, 7, 5, 7, 7, 7, 7, 7

Offset: 1

Tilman Neumann, Oct 04 2008, Oct 06 2008

The triangle T(n,k) contains many zeros. The distribution of nonzero entries is quite chaotic, but shows regular patterns, too, e.g.:
1) T(n,1) > 0 for n prime or n=4; T(n,1)=0 else
2) T(5k,k) > 0 for all k
More generally, it seems that:
3) T(pk,k) > 0 for k>0 and primes p
The following table depicts the zero (-) and nonzero (x) entries for the first 80 rows of the triangle:
-
xx
x-x
xxxx
x---x
-xxxxx
x-----x
-xxx-xxx
--x-x-x-x
-x--xx--xx
x---------x
---xxxxxxxxx
x-----------x
-x----xxxxxxxx
--x-x-x-x-x-x-x
-----xxx-x-x-xxx
x---------------x
-----x-xxx-x-x-xxx
x-----------------x
---x---xxxxx-x-xxxxx
--x---x-x---x-x---x-x
-x--------xxxx----xxxx
x---------------------x
-------x-xxx-xxx-xxx-xxx
----x---x---x---x---x---x
-x----------xx--xx--xx--xx
--------x-x-x-x-x-x-x-x-x-x
---x-----x--xxxxxxxxxxxxxxxx
x---------------------------x
-----x---x-x--xxxxxxxxxxxxxxxx
x-----------------------------x
-------------xxx-x-x-x-x-x-x-xxx
--x-------x-x-x-------x-----x-x-x
-x--------------xx--------------xx
----x-x---x---x-x-----x---x-x-x---x
-----------x-x-xxxxx---x-x-x-x-xxxxx
x-----------------------------------x
-x----------------xxxx------------xxxx
--x---------x-x---x-x-----x---x-x---x-x
-------x---x---x-xxx-xxx---x-x-x-xxx-xxx
x---------------------------------------x
-----x-----x-x-x-x-xxx-xxx---x-x-x-xxx-xxx
x-----------------------------------------x
---x---------x------xxxxxxxx-x-x-x-xxxxxxxxx
--------x---x-x-x-x-x-x-x-x-x---x-x-x-x-x-x-x
-x--------------------xxxxxxxx--------xxxxxxxx
x---------------------------------------------x
---------------x-x---xxx-x-x-xxx-x-x--xx-x-x-xxx
------x-----x-----x-----x-----x-----x-----x-----x
---------x---x---x---x--xx---x--xx---x--xx---x--xx
--x-------------x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x
---x-----------x--------xxxx-x-xxxxx---xxxxx-x-xxxxx
x---------------------------------------------------x
-----------------x-x-x-x-xxxxx-x-xxxxx-x-xxxxx-x-xxxxx
----x-----x---x---------x-----x---x---------x-----x---x
-------x-----x-----------xxx-xxx--xx-xxx-xxx-xxx-xxx-xxx
--x---------------x-x---------------x-x---------------x-x
-x--------------------------xx--xx--xx--xx--xx--xx--xx--xx
x---------------------------------------------------------x
-----------x---x---x-x-x----xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x-----------------------------------------------------------x
-x----------------------------xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
--------x-----x-----x-x-x-x-----x-----x-x-x-x-----x-----x-x-x-x
-----------------------------xxx-x-x-x-x-x-x-x-x-x-x-x-x-x-x-xxx
----x-------x---x---x---x---x---x---x---x---x-------x---x---x---x
-----x---------x-----x-x-x-x-x--xx-x---x-x---x-x-------x-x-x---xxx
x-----------------------------------------------------------------x
---x---------------x------------xxxx-------------x-x------------xxxx
--x-------------------x-x-x-x-x-x-------x-x-x-x-x-x-------x-x-x-x-x-x
---------x---x-x-x---x---x-x-x---xxxxx---x---x---x-x-x---x---x-x-xxxxx
x---------------------------------------------------------------------x
-----------------------x-x-x-x-x-xxx-xxx-x-x-x-x-x-x-x-x---x-x-x-xxx-xxx
x-----------------------------------------------------------------------x
-x----------------------------------xx--xx--------------------------xx--xx
--------------x---x---x-x-x---x-x-x-x-x---x-x-x-x-x---x-x-x-x-x---x-x-x-x-x
---x-----------------x--------------xxxxxxxx---------x-x-x-x--------xxxxxxxx
------x---x-----x-----x---x-x-----x-x---------x-----x---x-x-----x-x---x-----x
-----x-----------x-------x-x-x-x-x-x-xxxxxxxxx-x-x-x-x-x-x-x-x-x-x-x-xxxxxxxxx
x-----------------------------------------------------------------------------x
---------------x---x---------------x-xxx-x-x-xxx---x---x-x-x-x-x---x-xxx-x-x-xxx
SUM(A057427(a(k)): 1<=k<=n) = A005127(n). - Reinhard Zumkeller, Jul 04 2009

			Triangle starts:
0;
1, 1;
2, 0, 1;
2, 3, 2, 1;
4, 0, 0, 0, 1;
0, 4, 3, 1, 3, 1;
6, 0, 0, 0, 0, 0, 1;
....

Cf. A000040, A008275, A061006 (first column).

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(stirling(n, k, 1) % n, ", ");); print(););} \\ Michel Marcus, Aug 10 2015

T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

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

A285982 a(n) = n! (mod n + 3).

Original entry on oeis.org

1, 1, 2, 0, 3, 0, 0, 0, 5, 0, 6, 0, 0, 0, 8, 0, 9, 0, 0, 0, 11, 0, 0, 0, 0, 0, 14, 0, 15, 0, 0, 0, 0, 0, 18, 0, 0, 0, 20, 0, 21, 0, 0, 0, 23, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 29, 0, 30, 0, 0, 0, 0, 0, 33, 0, 0, 0, 35, 0, 36, 0, 0, 0, 0, 0, 39, 0, 0, 0, 41, 0, 0

Offset: 0

Seiichi Manyama, Apr 29 2017

nonn

Nonzero terms are a(2 * A130290(n) - 2) = A130290(n) for n > 1. - David A. Corneth, Apr 30 2017

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Wilson's theorem

Cf. A000142, A061006, A130290.

seq(n! mod (n+3), n=0..100); # Robert Israel, Apr 30 2017

a(n) = n! % (n+3); \\ Michel Marcus, Apr 30 2017

a(n) = A000142(n) (mod n + 3).

If n > 1 and a(n) > 0, a(n) = n/2 + 1 and n + 3 is a prime.

A180589 a(n) = floor(n!*h(n)/n), where h(n) = Sum_{k=1..n} 1/k.

Original entry on oeis.org

1, 1, 3, 12, 54, 294, 1866, 13698, 114064, 1062864, 10958530, 123870240, 1523289156, 20247546240, 289277533440, 4420892649600, 71965034739952, 1243166003251200, 22713955095665178, 437647401838080000, 8868800513341440000, 188567126333429760000, 4197346376195350706086

Offset: 1

Gary Detlefs, Sep 10 2010

nonn

Cf. A000254, A061006.

h:= n->sum(1/k,k=1..n):seq(floor(n!*h(n)/n),n=1..25);

a[n_]:=Floor[n!*Sum[1/k,{k,n}]/n]; Array[a,23] (* Stefano Spezia, Apr 20 2025 *)

a(n) = (A000254(n) - A061006(n))/n.

a(21)-a(23) from Stefano Spezia, Apr 20 2025

cp's OEIS Frontend

A061007 a(n) = -(n-1)! mod n.

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A061008 a(n) = Sum_{j=1..n} (-(n-1)! mod n).

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A061009 a(n) = -2 + Sum_{j=1..n} (-(n-1)!) mod n.

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A119688 a(n) = n!! mod (n+1).

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A238002 Count with multiplicity of prime factors of n in (n - 1)!.

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A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

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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.

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