A039716 -id:A039716 - cp's OEIS frontend

A063414 Number of digits in prime(n)! (A039716).

1, 1, 3, 4, 8, 10, 15, 18, 23, 31, 34, 44, 50, 53, 60, 70, 81, 84, 95, 102, 106, 117, 125, 137, 152, 160, 164, 173, 177, 185, 214, 222, 235, 239, 261, 265, 279, 292, 301, 314, 328, 332, 355, 359, 369, 373, 401, 429, 438, 443, 452, 467, 471, 495, 510, 524, 539

Offset: 1

Jason Earls, Oct 06 2001

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A039716.

[seq(floor(log10(factorial(ithprime(i))))+1, i = 1 .. 100)]; # César Eliud Lozada, Nov 23 2014

a(n)=#digits(prime(n)!); \\ Joerg Arndt, Nov 23 2014

A134557 Triangle multiplicatively decoded from A039716, the factorials of the prime numbers, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 10, 5, 2, 1, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 8, 3, 2, 1, 1, 1, 1, 19, 9, 4, 3, 2, 1, 1, 1, 1, 25, 13, 6, 4, 2, 2, 1, 1, 1, 1, 26, 14, 7, 4, 2, 2, 1, 1, 1, 1, 1, 34, 17, 8, 5, 3, 2, 2, 1, 1, 1, 1, 1, 38, 18, 9, 5, 3, 3, 2, 2, 1, 1, 1, 1, 1, 39, 19, 9, 6, 3, 3, 2

Offset: 1

Alonso del Arte, Jan 22 2008

			Triangle begins:
{1}
{1, 1}
{3, 1, 1}
{4, 2, 1, 1}
{8, 4, 2, 1, 1}
{10, 5, 2, 1, 1, 1}
{15, 6, 3, 2, 1, 1, 1}
{16, 8, 3, 2, 1, 1, 1, 1}
Row 5 of the triangle is 8, 4, 2, 1, 1 because A039716(5) = 39916800 and 39916800 = 2^8 * 3^4 * 5^2 * 7^1 * 11^1.

N. J. A. Sloane, The Encyclopedia of Integer Sequences. New York: Academic Press (1995): Fig. M1722, "Multiplicative encoding of a triangular array"

PlanetMath, Multiplicative encoding

Cf. A039716.

ColumnForm[Table[Take[Flatten[FactorInteger[Prime[n]! ]], {2, 2n, 2}], {n, 15}], Center]

A139159 a(n) = prime(n)! + 1.

Original entry on oeis.org

3, 7, 121, 5041, 39916801, 6227020801, 355687428096001, 121645100408832001, 25852016738884976640001, 8841761993739701954543616000001, 8222838654177922817725562880000001, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001

Offset: 1

Artur Jasinski, Apr 11 2008

Cf. A039716.
Cf. A093804, A103319.
Cf. A139160, A139161, A139162, A139163, A139164, A139165, A139166, A139089, A139168.

Prime[Range[12]]!+1 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)!+1 \\ Charles R Greathouse IV, Apr 29 2015

a(n) = A039716(n) + 1. - Michel Marcus, Nov 08 2013

More terms from Michel Marcus, Aug 10 2025

A139189 a(n) = prime(n)!-1.

Original entry on oeis.org

1, 5, 119, 5039, 39916799, 6227020799, 355687428095999, 121645100408831999, 25852016738884976639999, 8841761993739701954543615999999, 8222838654177922817725562879999999, 13763753091226345046315979581580902399999999, 33452526613163807108170062053440751665151999999999, 60415263063373835637355132068513997507264511999999999, 258623241511168180642964355153611979969197632389119999999999, 4274883284060025564298013753389399649690343788366813724671999999999999, 138683118545689835737939019720389406345902876772687432540821294940159999999999999

Offset: 1

Artur Jasinski, Apr 11 2008

nonn

Numbers of the form (prime(n)!-m)/m :
for m=1 see A139189
for m=2 see A139190
for m=3 see A139191
for m=4 see A139192
for m=5 see A139193
for m=6 see A139194
for m=7 see A139195
for m=8 see A139196
for m=9 see A139197
for m=10 see A139198

Alois P. Heinz, Table of n, a(n) for n = 1..85

Cf. A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198.

[(Factorial(NthPrime(n))-1): n in [1..20]]; // Vincenzo Librandi, Jun 02 2017

Mathematica
```
Table[(Prime[n]! - 1)/1, {n, 1, 20}]
```

a(n) = prime(n)!-1 \\ Charles R Greathouse IV, Apr 29 2015

a(n)=A039716(n)-1. - R. J. Mathar, May 25 2008

More terms from R. J. Mathar, May 25 2008

A139193 Natural numbers of the form (prime(n)! - 5)/5.

Original entry on oeis.org

23, 1007, 7983359, 1245404159, 71137485619199, 24329020081766399, 5170403347776995327999, 1768352398747940390908723199999, 1644567730835584563545112575999999

Offset: 1

Artur Jasinski, Apr 11 2008

nonn

Cf. A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198.

A039716 := proc(n) factorial(ithprime(n)) ; end: A139193 := proc(n) A039716(n)/5-1 ; end: seq(A139193(n),n=3..14) ; # R. J. Mathar, Sep 18 2009

Mathematica
```
Table[(Prime[n]! - 5)/5, {n, 3, 20}]
```

A139196 a(n) = (prime(n)!-8)/8.

Original entry on oeis.org

14, 629, 4989599, 778377599, 44460928511999, 15205637551103999, 3231502092360622079999, 1105220249217462744317951999999, 1027854831772240352215695359999999, 1720469136403293130789497447697612799999999

Offset: 3

Artur Jasinski, Apr 11 2008

nonn

Cf. A039716, A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198.

Mathematica
```
Table[(Prime[n]! - 8)/8, {n, 3, 20}]
```

a(n) = (A039716(n) - 8)/8. - Elmo R. Oliveira, Jan 20 2023

Edited by Max Alekseyev, Sep 15 2009

A139198 a(n) = prime(n)!/10 - 1.

Original entry on oeis.org

11, 503, 3991679, 622702079, 35568742809599, 12164510040883199, 2585201673888497663999, 884176199373970195454361599999, 822283865417792281772556287999999

Offset: 3

Artur Jasinski, Apr 11 2008

Cf. A039716, A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197.

Mathematica
```
Table[(Prime[n]! - 10)/10, {n, 3, 20}]
```

a(n) = prime(n)!/10 - 1 \\ David A. Corneth, Jun 02 2017

a(n) = A039716(n)/10 - 1. - Elmo R. Oliveira, Jan 20 2023

A103855 a(n) = prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 91, 4831, 39914491, 6226990771, 355687427585491, 121645100399132311, 25852016738884753547131, 8841761993739701954537146306771, 8222838654177922817725362319509871, 13763753091226345046315979581573481661865191, 33452526613163807108170062053440751360901736472791

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..86
Reinhard Zumkeller, p(n)! - p(n)# + 1.

Cf. A103856, A103857, A103858, A103859, A103860, A103861, A103890.

Cf. A002110, A039716, A092435.

primorial[n_] := Product[Prime[i], {i, n}]; A103855[n_] := Prime[n]! - primorial[n] + 1; Array[A103855, 20] (* G. C. Greubel, May 09 2017 *)
With[{nn=15},#[[1]]-#[[2]]+1&/@Thread[{Prime[Range[nn]]!,FoldList[Times,Prime[Range[nn]]]}]] (* Harvey P. Dale, Aug 11 2025 *)

a(n) = A039716(n) - A002110(n) + 1 = A002110(n) * (A092435(n) - 1) + 1.

A103890 a(n) = prime(n)! / prime(n)# + 1.

Original entry on oeis.org

2, 2, 5, 25, 17281, 207361, 696729601, 12541132801, 115880067072001, 1366643159020339200001, 40999294770610176000001, 1854768736099424576471040000001, 109950690675973888893203251200000001, 4617929008390903333514536550400000001

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

Reinhard Zumkeller, p(n)! / p(n)# + 1.

Cf. A103855, A103891, A103892, A103893, A103894, A103895, A103896.

#[[1]]/#[[2]]&/@With[{nn=15},Thread[{Prime[Range[nn]]!,FoldList[ Times,Prime[ Range[nn]]]}]]+1 (* Harvey P. Dale, May 21 2019 *)

a(n) = prime(n)!/vecprod(primes(n)) + 1; \\ Michel Marcus, Nov 12 2023

a(n) = A039716(n)/A002110(n) + 1 = A092435(n) + 1.

A092435 Prime factorials divided by their corresponding primorials.

Original entry on oeis.org

1, 1, 4, 24, 17280, 207360, 696729600, 12541132800, 115880067072000, 1366643159020339200000, 40999294770610176000000, 1854768736099424576471040000000, 109950690675973888893203251200000000, 4617929008390903333514536550400000000, 420600974084243475616503989010432000000000

Offset: 1

Don Willard (dwillard(AT)prairie.cc.il.us), Mar 23 2004

nonn

			E.g., 2 factorial divided by 2 primorial is 1; 3 factorial is 6, divided by 3 primorial (3*2=6) is also 1; 5 factorial is 120, divided by 5 primorial (5*3*2=30) is 4 and so forth.

Subsequence of A036691. - Chayim Lowen, Jul 23 2015

Cf. A002110, A039716.

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1)*mul(i, i=ithprime(n-1)+1..ithprime(n)-1))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Jan 15 2025

Table[ Prime[n]! / Times @@ Prime[ Range[ n]], {n, 13}] (* Robert G. Wilson v, Mar 25 2004 *)

a(n)=prime(n)!/prod(i=1,n,prime(i)) \\ Ralf Stephan, Dec 21 2013

p!/p# = A039716/A002110.
Partial products of A061214. - Lekraj Beedassy, Nov 06 2006
From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A036691(A065890(n)).
a(n) = A000142(A002808(A065890(n)))/A034386(A002808(A065890(n))).
a(n) = Product_{k=1..n} prime(k)^(A085604(prime(n),k)-1).
a(n) = A049614(prime(n)).
a(n) = Product_{k=1..prime(n)} k^A066247(k). (End)

Edited by Robert G. Wilson v, Mar 25 2004

More terms from Michel Marcus, Jan 15 2025

cp's OEIS Frontend

A063414 Number of digits in prime(n)! (A039716).

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A134557 Triangle multiplicatively decoded from A039716, the factorials of the prime numbers, read by rows.

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A139159 a(n) = prime(n)! + 1.

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A139189 a(n) = prime(n)!-1.

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A139193 Natural numbers of the form (prime(n)! - 5)/5.

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A139196 a(n) = (prime(n)!-8)/8.

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A139198 a(n) = prime(n)!/10 - 1.

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A103855 a(n) = prime(n)! - prime(n)# + 1.

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A103890 a(n) = prime(n)! / prime(n)# + 1.

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