Displaying 1-10 of 24 results found.
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
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
PROG
(Magma) [n: n in [0..500] | IsPrime((Factorial(n)+3) div 3)]; // Vincenzo Librandi, Dec 12 2011
CROSSREFS
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).
EXTENSIONS
1330 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Numbers n such that (n! + 2)/2 is a prime.
+10
56
2, 4, 5, 7, 8, 13, 16, 30, 43, 49, 91, 119, 213, 1380, 1637, 2258, 4647, 9701, 12258
PROG
(PARI) \\ x such that (x!+2)/2 is prime
xfactpk(n, k=2) = { for(x=2, n, y = (x!+k)/k; if(isprime(y), print1(x, ", ")) ) }
(Magma) [ n: n in [1..300] | IsPrime((Factorial(n)+2) div 2) ];
CROSSREFS
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).
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
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
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
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
EXTENSIONS
Definition corrected by Derek Orr, Mar 28 2014
Numbers k for which (9 + k!)/9 is prime.
+10
50
8, 46, 87, 168, 259, 262, 292, 329, 446, 1056, 3562, 11819, 26737
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
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
CROSSREFS
Cf. A139068 (primes of the form (9 + k!)/9).
EXTENSIONS
a(11) from Dimitris Zygiridis (dmzyg70(AT)gmail.com), Jul 25 2008
Primes of the form (8+k!)/8.
+10
41
631, 45361, 453601, 59875201, 10897286401, 304112751022080001, 3231502092360622080001, 77556050216654929920001, 1105220249217462744317952000001, 332283946848556096005453226376826986289954816000000001
COMMENTS
For numbers k for which (8+k!)/8 is prime see A151913.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 8)/8], AppendTo[a, (n! + 8)/8]], {n, 1, 50}]; a
CROSSREFS
Cf. A020458, A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139155, A151913.
EXTENSIONS
Corrected link to sequence of indexes. - Serge Batalov, Feb 17 2015
Primes of the form k!/9 + 1.
+10
35
4481, 611402462201343216650033936533361654773516861440000000001, 234195255375503079690400057633265510581087082006817356924774723468294901747510352675631491470712754833859385753600000000000000000001
COMMENTS
For numbers k for which (9+k!)/9 is prime see A137390.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 9)/9], AppendTo[a, [(n! + 9)/9]], {n, 1, 150}]; a
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.
Primes of the form (10+k!)/10.
+10
34
13, 73, 3991681, 47900161, 130767436801, 2585201673888497664001, 40329146112660563558400001, 1376375309122634504631597958158090240000001, 11962222086548019456196316149565771506438373376000000001
COMMENTS
For numbers k for which (10+k!)/10 is prime see A139071.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 10)/10], AppendTo[a, [(n! + 10)/10]], {n, 1, 50}]; a
CROSSREFS
Cf. A020458, A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139062, A139063, A139064, A139065, A139066, A139068, A137390, A139071, A139072, A139157.
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
COMMENTS
For primes of the form (5+n!)/5 see A139059.
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. 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).
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
COMMENTS
For primes of the form (4+k!)/4, see A139060.
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 *)
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).
Numbers k for which (6+k!)/6 is prime.
+10
25
3, 4, 10, 11, 13, 14, 17, 21, 82, 115, 165, 167, 173, 174, 208, 225, 380, 655, 1187, 2000, 2568, 3010, 4542, 8750, 12257, 12601, 24083
COMMENTS
For primes of the form (6+k!)/6, see A139062.
MATHEMATICA
a = {}; Do[If[PrimeQ[(n! + 6)/6], AppendTo[a, n]], {n, 1, 500}]; a
CROSSREFS
Cf. A082672, A089085, A089130, A117141, A007749, A139056, A139057, A139058, A139059, A139060, A139061, A139061, A139062, A139063, A139064, A139065, A139066, A020458.
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).
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