A082672 -id:A082672

A089085

Numbers k such that (k! + 3)/3 is prime.

+10
57

3, 5, 6, 8, 11, 17, 23, 36, 77, 93, 94, 109, 304, 497, 1330, 1996, 3027, 3053, 4529, 5841, 20556, 26558, 28167

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(21) > 20000. The PFGW program has been used to certify all the terms up to a(20), using the "N-1" deterministic test. - Giovanni Resta, Mar 31 2014

LINKS

Table of n, a(n) for n=1..23.

MATHEMATICA

Select[Range[0, 1400], PrimeQ[(#!+3)/3] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

PROG

(Magma) [n: n in [0..500] | IsPrime((Factorial(n)+3) div 3)]; // Vincenzo Librandi, Dec 12 2011

(PARI) is(n)=ispseudoprime(n!\3+1) \\ Charles R Greathouse IV, Mar 21 2013

CROSSREFS

Cf. A089131.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

KEYWORD

nonn

AUTHOR

Cino Hilliard, Dec 05 2003

EXTENSIONS

More terms from Don Reble, Dec 06 2003

1330 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

Typo in Mma program corrected by Vincenzo Librandi, Dec 12 2011

a(16)-a(20) from Giovanni Resta, Mar 31 2014

a(21)-a(23) from Serge Batalov, Feb 17 2015

STATUS

approved

A139056

Numbers k for which (k!-3)/3 is prime.

+10
56

4, 6, 12, 16, 29, 34, 43, 111, 137, 181, 528, 2685, 39477, 43697

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Corresponding primes (k!-3)/3 are in A139057.

a(13) > 10000. The PFGW program has been used to certify all the terms up to a(12), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Mar 28 2014

98166 is a member of the sequence but its index is not yet determined. The interval where sieving and tests were not run is [60000,90000]. - Serge Batalov, Feb 24 2015

LINKS

Table of n, a(n) for n=1..14.

C. Caldwell. The Prime database entry for the prime generated by a(i)=98166.

MATHEMATICA

a = {}; Do[If[PrimeQ[(-3 + n!)/3], AppendTo[a, n]], {n, 1, 1000}]; a

PROG

(PARI) for(n=1, 1000, if(floor(n!/3-1)==n!/3-1, if(ispseudoprime(n!/3-1), print(n)))) \\ Derek Orr, Mar 28 2014

CROSSREFS

Cf. A007749, A117141.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205.

Cf. n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071.

Cf. m*n!-1 is a prime: A076133, A076134, A099350, A099351, A180627-A180631.

Cf. m*n!+1 is a prime: A051915, A076679-A076683, A178488, A180626, A126896.

KEYWORD

nonn,more

AUTHOR

Artur Jasinski, Apr 07 2008

EXTENSIONS

Definition corrected by Derek Orr, Mar 28 2014

a(8)-a(11) from Derek Orr, Mar 28 2014

a(12) from Giovanni Resta, Mar 28 2014

a(13)-a(14) from Serge Batalov, Feb 24 2015

STATUS

approved

A137390

Numbers k for which (9 + k!)/9 is prime.

+10
50

8, 46, 87, 168, 259, 262, 292, 329, 446, 1056, 3562, 11819, 26737

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

No other k exists, for k <= 6000. - Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008

The next number in the sequence, if one exists, is greater than 10944. - Robert Price, Mar 16 2010

Borrowing from A139074 another term in this sequence is 26737. There may be others between 10944 and 26737. - Robert Price, Dec 13 2011

There are no other terms for k < 26738. - Robert Price, Feb 10 2012

LINKS

Table of n, a(n) for n=1..13.

EXAMPLE

a(11) = 3562 because 3562 is the 11th natural number for which k!/9 + 1 is prime. 3562 is the new term.

MATHEMATICA

a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, n]], {n, 1, 500}]; a

PROG

(PARI) for(n=6, 1e4, if(ispseudoprime(n!/9+1), print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

(PFGW) ABC2 $a!/9+1

a: from 6 to 1000 // Jinyuan Wang, Feb 04 2020

CROSSREFS

Cf. A139068 (primes of the form (9 + k!)/9).

Cf. k!/m - 1 is a prime: A002982, A082671, A139056, A139199-A139205.

Cf. (m + k!)/m is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A139071.

KEYWORD

nonn,more

AUTHOR

Artur Jasinski, Apr 09 2008

EXTENSIONS

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

a(10) corrected from 1053 to 1056 by Dmitry Kamenetsky, Jul 12 2008

a(11) from Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008

a(12)-a(13) from Robert Price, Feb 10 2012

STATUS

approved

A089130

Primes of the form (k! + 2)/2.

+10
43

2, 13, 61, 2521, 20161, 3113510401, 10461394944001, 132626429906095529318154240000001, 30207631531686917818677566034256998753632256000000001

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..13

FORMULA

a(n) = A139149(A082672(n)). - Amiram Eldar, Oct 13 2024

MATHEMATICA

Select[Table[(n!+2)/2, {n, 0, 100}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

PROG

(PARI) nfactp2d2(n) = { for(x=1, n, y=floor((x!+ 2)/2); if(isprime(y), print1(y", ")) ) }

(Magma) [ a: n in [1..50] | IsPrime(a) where a is (Factorial(n)+2) div 2 ];

CROSSREFS

Cf. A082672, A139149.

KEYWORD

nonn

AUTHOR

Cino Hilliard, Dec 05 2003

STATUS

approved

A139066

Primes of the form (8+k!)/8.

+10
41

631, 45361, 453601, 59875201, 10897286401, 304112751022080001, 3231502092360622080001, 77556050216654929920001, 1105220249217462744317952000001, 332283946848556096005453226376826986289954816000000001

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For numbers k for which (8+k!)/8 is prime see A151913.

The next term (a(11)) has 174 digits. - Harvey P. Dale, May 10 2016

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..11

FORMULA

a(n) = A139155(A151913(n)). - Amiram Eldar, Oct 14 2024

MATHEMATICA

a = {}; Do[If[PrimeQ[(n! + 8)/8], AppendTo[a, (n! + 8)/8]], {n, 1, 50}]; a

Select[(8+Range[50]!)/8, PrimeQ] (* Harvey P. Dale, May 10 2016 *)

PROG

(PARI) for(k=4, 1e3, if(ispseudoprime(t=k!/8+1), print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

CROSSREFS

Cf. A020458, A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139155, A151913.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Apr 07 2008

EXTENSIONS

Corrected link to sequence of indexes. - Serge Batalov, Feb 17 2015

a(10) from Harvey P. Dale, May 10 2016

STATUS

approved

A139068

Primes of the form k!/9 + 1.

+10
35

4481, 611402462201343216650033936533361654773516861440000000001, 234195255375503079690400057633265510581087082006817356924774723468294901747510352675631491470712754833859385753600000000000000000001

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For numbers k for which (9+k!)/9 is prime see A137390.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..9

FORMULA

a(n) = A139156(A137390(n)). - Amiram Eldar, Oct 14 2024

MATHEMATICA

a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, [(n! + 9)/9]], {n, 1, 150}]; a

Select[Range[100]!/9+1, PrimeQ] (* Harvey P. Dale, Aug 17 2017 *)

PROG

(PARI) for(n=6, 1e4, if(ispseudoprime(t=n!/9+1), print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

CROSSREFS

Cf. A020458, A007749, A082672, A089085, A089130, A117141, A137390, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139066, A137390, A139070, A139071, A139072, A139156.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Apr 07 2008

STATUS

approved

A139070

Primes of the form (10+k!)/10.

+10
34

13, 73, 3991681, 47900161, 130767436801, 2585201673888497664001, 40329146112660563558400001, 1376375309122634504631597958158090240000001, 11962222086548019456196316149565771506438373376000000001

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For numbers k for which (10+k!)/10 is prime see A139071.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..16

FORMULA

a(n) = A139157(A139071(n)). - Amiram Eldar, Oct 14 2024

MATHEMATICA

a = {}; Do[If[PrimeQ[(n! + 10)/10], AppendTo[a, [(n! + 10)/10]], {n, 1, 50}]; a

Select[(Range[50]!+10)/10, PrimeQ] (* Harvey P. Dale, Sep 18 2013 *)

PROG

(PARI) for(k=5, 1e3, if(ispseudoprime(t=k!/10+1), print1(t", "))) \\ Charles R Greathouse IV, Jul 15 2011

CROSSREFS

Cf. A020458, A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139066, A139068, A137390, A139071, A139072, A139157.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Apr 07 2008

STATUS

approved

A139075

Primes p arising in A139074.

+10
30

3, 2, 3, 31, 1009, 2, 5702401, 631

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(23) = (23+1579!)/23. - Andrew V. Sutherland, Apr 11, 2008.

Smallest mother factorial prime p of order n, i.e. smallest prime of the form (p!+n)/n where p is prime.

For smallest daughter factorial prime p of order n see A139074.

For smallest father factorial prime p of order n see A139207.

For smallest son factorial prime p of order n see A139206.

a(9)=26737!/9+1 is a 106758 digit (probable) prime. Easily calculated but too large to enter here a(10)=13, a(11)=566092801, a(12)=11. [Robert Price, Jan 19 2011]

LINKS

Table of n, a(n) for n=1..8.

MATHEMATICA

a = {}; Do[k = 1; While[ ! PrimeQ[(Prime[k]! + n)/n], k++ ]; AppendTo[a, Prime[(Prime[k]! + n)/n]], {n, 1, 8}]; a

CROSSREFS

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A136019, A136020, A136026, A136027.

KEYWORD

nonn

AUTHOR

Artur Jasinski, Apr 08 2008, Apr 21 2008

STATUS

approved

A139058

Numbers n such that (5+n!)/5 is prime.

+10
26

7, 9, 11, 14, 19, 23, 45, 121, 131, 194, 735, 751, 1316, 1372, 2084, 2562, 5678, 5758, 12533, 24222

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For primes of the form (5+n!)/5 see A139059.

a(21) > 25000. - Robert Price, Nov 20 2016

LINKS

Table of n, a(n) for n=1..20.

MATHEMATICA

a = {}; Do[If[PrimeQ[(n! + 5)/5], AppendTo[a, n]], {n, 1, 751}]; a

PROG

(Magma) [ n: n in [5..734] | IsPrime((Factorial(n)+5) div 5) ];

(PARI) A139058(n) = local(k=(n!+5)\5); if(isprime(k), k, 0);

for(n=5, 800, if(A139058(n)>0, print1(n, ", ")))

CROSSREFS

Cf. A139059.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

KEYWORD

nonn

AUTHOR

Artur Jasinski, Apr 07 2008

EXTENSIONS

More terms from Serge Batalov, Feb 18 2015

a(19)-a(20) from Robert Price, Nov 20 2016

STATUS

approved

A139061

Numbers n for which (4+n!)/4 is prime.

+10
25

4, 5, 6, 13, 21, 25, 32, 40, 61, 97, 147, 324, 325, 348, 369, 1290, 1342, 3167, 6612, 8176, 10990

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For primes of the form (4+k!)/4, see A139060.

a(22) > 25000. - Robert Price, Jan 10 2017

LINKS

Table of n, a(n) for n=1..21.

MATHEMATICA

a = {}; Do[If[PrimeQ[(n! + 4)/4], AppendTo[a, n]], {n, 1, 500}]; a

Select[Range[500], PrimeQ[(4+#!)/4]&] (* Harvey P. Dale, Mar 24 2011 *)

PROG

(PARI) for(n=4, 1e3, if(ispseudoprime(n!/4+1), print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

CROSSREFS

Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458, A139068, A137390, A139070, A139071, A139072.

Cf. n!/m-1 is a prime: A002982, A082671, A139056, A139199-A139205; n!/m+1 is a prime: A002981, A082672, A089085, A139061, A139058, A139063, A139065, A151913, A137390, A139071 (1<=m<=10).

KEYWORD

nonn

AUTHOR

Artur Jasinski, Apr 07 2008

EXTENSIONS

More terms from Serge Batalov, Feb 18 2015

a(19) - a(21) from Robert Price, Jan 10 2017

STATUS

approved