A006990 -id:A006990 - cp's OEIS frontend

A297707 a(n) = Product_{k=1..n-1} n!k, where n!k is k-tuple factorial of n.

1, 2, 18, 768, 90000, 44789760, 30494620800, 121762322841600, 393644011735296000, 5618427494400000000000, 107587910030480590233600000, 5951222311476064581656248320000, 176804782652901880753915871232000000, 69819090744423637487544223697731584000000

Offset: 1

Lechoslaw Ratajczak, Jan 03 2018

nonn

What is the least n > 2 for which a(n) - prevprime(a(n)) is a composite number? If such a number n exists, it is greater than 250.

The least n for which nextprime(a(n)) - a(n) is a composite number is 158.

			a(2) = (2!1) = (2*1) = 2;
a(3) = (3!1)*(3!2) = (3*2*1)*(3*1) = 18;
a(4) = (4!1)*(4!2)*(4!3) = (4*3*2*1)*(4*2)*(4*1) = 768;
a(5) = (5!1)*(5!2)*(5!3)*(5!4) = (5*4*3*2*1)*(5*3*1)*(5*2)*(5*1) = 90000.

Michel Marcus, Table of n, a(n) for n = 1..100

Cf. A000142, A006882, A007661, A007662, A085157, A085158, A114799, A114800.

Cf. A114806, A288327, A006990, A033933.

b:= proc(n, k) option remember; `if`(n<1, 1, n*b(n-k, k)) end:
a:= n-> mul(b(n, k), k=1..n-1):
seq(a(n), n=1..20);  # Alois P. Heinz, Dec 02 2018

Array[(#^(# - 1)) Product[k^DivisorSigma[0, # - k], {k, # - 1}] &, 13] (* Michael De Vlieger, Jan 04 2018 *)

a(n) = (n^(n-1))*prod(k=1, n-1, k^numdiv(n-k)); \\ Michel Marcus, Dec 02 2018

a(n) = Product_{t=1..n-1} (Product_{k=0..floor((n-1)/t)} (n-t*k)).

a(n) = (n^(n-1))*Product_{k=1..n-1} k^tau(n-k).

A340013 The prime gap, divided by two, which surrounds n!.

Original entry on oeis.org

1, 3, 7, 4, 6, 27, 15, 11, 7, 15, 45, 10, 45, 38, 45, 39, 95, 30, 31, 52, 93, 102, 95, 48, 22, 84, 127, 54, 94, 40, 19, 145, 87, 129, 49, 22, 85, 68, 66, 88, 90, 78, 146, 95, 156, 78, 71, 79, 225, 60, 65, 175, 66, 305, 192, 196, 215, 205, 420, 101, 186, 213, 160

Offset: 3

Robert G. Wilson v, Jan 09 2021

nonn

A theorem states that between (n+1)! + 2 and (n+1)! + (n+1) inclusive, there are n consecutive composite integers, namely 2, 3, 4, ..., n, n+1.

Records: 1, 3, 7, 27, 45, 95, 102, 127, 145, 146, 156, 225, 305, 420, 804, 844, 1173, 1671, 1725, 1827, 2570, 2930, 3318, 5142, 5946, 6837, 7007, 8208, 10221, ..., .

			For a(1), there are no positive primes which surround 1!. Therefore a(1) is undefined.
For a(2), there are two contiguous primes {2, 3} with 2 being 2!. The prime gap is 1. However, the two primes do not surround 2!, so a(2) is undefined.
For a(3), the following set of numbers, {5, 6, 7}, with 3! being in the middle. The prime gap is 2; therefore, a(3) = 1.
For a(4), the following set of numbers, {23, 24, 25, 26, 27, 28, 29} with 4! in between the two primes 23 & 29. The prime gap is 6, so a(4) = 3.

Robert Israel, Table of n, a(n) for n = 3..607

Cf. A000142, A002981, A006990, A033932, A033933, A037151, A054588, A058054, A073308.

a:= n-> (f-> (nextprime(f-1)-prevprime(f+1))/2)(n!):
seq(a(n), n=3..70);  # Alois P. Heinz, Jan 09 2021

a[n_] := (NextPrime[n!, 1] - NextPrime[n!, -1])/2; Array[a, 70, 3]

a(n) = (nextprime(n!+1) - precprime(n!-1))/2; \\ Michel Marcus, Jan 11 2021

from sympy import factorial, nextprime, prevprime
def A340013(n):
    f = factorial(n)
    return (nextprime(f)-prevprime(f))//2 # Chai Wah Wu, Jan 23 2021

a(n) = (A037151(n) - A006990(n))/2 = (A033932(n) + A033933(n))/2.

a(n) = A054588(n)/2 = A058054(n)/2. - Alois P. Heinz, Jan 09 2021

A053712 Lower balancing primes to prime-balanced factorials.

Original entry on oeis.org

5, 113, 3628789, 51090942171709439969

Offset: 1

Labos Elemer, Feb 10 2000

nonn

The next two terms are 171!-397 and 190!-409. - Jud McCranie, Jul 04 2000

			113 is balancing 5! = 120 from below, where 5! = 120 is a balanced factorial.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A037151, A006990, A006562, A053709, A053710, A053711, A053713.

a(n) = A053709(n)! - A053711(n) = A053710(n) - A053711(n). - Amiram Eldar, Mar 10 2025

A053713 Upper balancing primes to prime-balanced factorials.

Original entry on oeis.org

7, 127, 3628811, 51090942171709440031

Offset: 1

Labos Elemer, Feb 10 2000

nonn

The next two terms are 171!+397 and 190!+409, which are too large to include. - Jud McCranie, Jul 04 2000

			127 is balancing 5! = 120 from above, where 5! = 120 is a balanced factorial.

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A033932, A033933, A037151, A006990, A006562, A053709, A053710, A053711, A053712.

a(n) = A053709(n)! + A053711(n) = A053710(n) + A053711(n). - Amiram Eldar, Mar 10 2025

A063959 Sum of the primes from 1 to n!.

Original entry on oeis.org

0, 0, 2, 10, 100, 1593, 41741, 1578242, 80294846, 5356015580, 451223209946, 46900682786541, 5891009442510166, 879657744587755114, 153967535281046615774, 31216213430872403460411, 7256556722488434503836458, 1917031284234466887065107947, 571083301099266868435687532291

Offset: 0

Jason Earls, Sep 03 2001

nonn

Sum of prime factors (without repetition) of (n!)!.

			a(4) = sum of primes <= 24. They are 2, 3, 5, 7, 11, 13, 17, 19 and 23. This sum is 100.

Amiram Eldar, Table of n, a(n) for n = 0..22 (extended using Kim Walisch's primesum; terms 0..20 from Daniel Suteu)
Kim Walisch, primesum program.

Cf. A000142, A034387, A006990, A294194.

NextPrim[n_] := (k = n + 1; While[ ! PrimeQ[k], k++ ]; k); s = 0; p = 1; Do[ Do[p = NextPrim[p]; s = s + p, {i, PrimePi[(n - 1)! ] + 1, PrimePi[(n)! ]}]; Print[s], {n, 1, 12} ]
Do[ Print[ Sum[ Prime[k], {k, 1, PrimePi[n! ]}]], {n, 0, 10} ]
Table[Total[Prime[Range[PrimePi[n!]]]],{n,0,9}] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Aug 17 2025 *)

sumprime(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s); for(n=0,22,print(sumprime(n!!)))

a(n) = A034387(A000142(n)). - Michel Marcus, Jun 14 2024

Better description and more terms from Robert G. Wilson v, Oct 04 2001
a(13)-a(15) from Donovan Johnson, May 03 2010
a(16)-a(18) from Daniel Suteu, Nov 15 2018

A053708 Nearest prime to n! (but not equal to n!).

Original entry on oeis.org

2, 3, 5, 23, 113, 719, 5039, 40343, 362867, 3628789, 39916801, 479001599, 6227020777, 87178291199, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029, 51090942171709439969, 1124000727777607680031, 25852016738884976639911

Offset: 1

Labos Elemer, Feb 10 2000

nonn

If n! is the average of its closest prime neighbors then the smaller prime is to be chosen (as in A051701).

			For 8! = 40320 the closest upper and lower primes are 40289 and 40343 with d = 31 and d = 23, so 40343 is closer to 8! than the lower neighbor.

Amiram Eldar, Table of n, a(n) for n = 1..449

Cf. A000142, A033932, A037151, A033933, A006990.

f[n_]:=Module[{nf=n!,s,l},s=NextPrime[nf,-1];l=NextPrime[nf];If[nf-s>l-nf,l,s]]
Table[f[i],{i,25}] (* Harvey P. Dale, Dec 08 2010 *)

Corrected by Rick L. Shepherd, Jan 11 2006

a(21)-a(23) from Amiram Eldar, Mar 10 2025

A178606 Smallest prime > n!! (double factorial of n).

Original entry on oeis.org

2, 3, 5, 11, 17, 53, 107, 389, 947, 3847, 10399, 46091, 135151, 645131, 2027033, 10321937, 34459429, 185794579, 654729139, 3715891217, 13749310577, 81749606417, 316234143227, 1961990553613, 7905853580633, 51011754393671

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 30 2010

nonn

Cf. A006990, A037151.

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; Table[PrimeNext[n!! ],{n,40}]

A178607 Largest prime < n!! (double factorial of n).

Original entry on oeis.org

2, 7, 13, 47, 103, 383, 941, 3833, 10391, 46073, 135131, 645097, 2027023, 10321919, 34459423, 185794463, 654729073, 3715891181, 13749310511, 81749606383, 316234143163, 1961990553581, 7905853580621, 51011754393599

Offset: 3

Vladimir Joseph Stephan Orlovsky, May 30 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 3..200

Cf. A006882, A006990, A037151, A178606.

PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k]; Table[PrimePrev[n!! ],{n,3,40}]
NextPrime[#,-1]&/@(Range[3,30]!!) (* Harvey P. Dale, Jul 13 2015 *)

a(n)=if(n<2, 1, n*a(n-2)); for(n=3,26,print1(precprime(a(n)-1)", ")); \\ Graziano Aglietti (mg5055(AT)mclink.it), Aug 23 2010

a(n) = A151799(A006882(n)). - R. J. Mathar, Jun 23 2010

Offset corrected by R. J. Mathar, Jun 23 2010

cp's OEIS Frontend

A297707 a(n) = Product_{k=1..n-1} n!k, where n!k is k-tuple factorial of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A340013 The prime gap, divided by two, which surrounds n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A053712 Lower balancing primes to prime-balanced factorials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A053713 Upper balancing primes to prime-balanced factorials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A063959 Sum of the primes from 1 to n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A053708 Nearest prime to n! (but not equal to n!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A178606 Smallest prime > n!! (double factorial of n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A178607 Largest prime < n!! (double factorial of n).

Views

Author