A139074 a(n) = smallest prime p such that p!/n + 1 is prime, or 0 if no such prime exists.

2, 2, 3, 5, 7, 3, 11, 7, 26737, 5, 13, 5

Offset: 1

Author

Artur Jasinski, Apr 08 2008, Apr 21 2008

Keywords

• more
• nonn

Comments

For the corresponding primes p see A139075.

a(9)>5000, a(13)>5000, a(22)>5000, a(23) = 1579. - Andrew V. Sutherland, Apr 21 2008, Apr 22 2008

a(10)=5, a(11)=13, a(12)=5

a(14)=17, a(15)=7, a(16)=13, a(17)=43, a(18)=7,

a(19)=31, a(20)=5, a(21)=7

a(24)=7, a(25)=47, a(26)=17, a(27)=17, a(28)=7,

a(29)=241, a(30)=5, a(31)=61, a(32)=11, a(33)=17,

a(34)=17, a(35)=29, a(36)=11, a(37)=61, a(38)=103,

a(39)=89, a(40)=7, a(41)=131, a(42)=11, a(43)=71,

a(44)=13, a(45)=7, a(46)=43, a(47)=73, a(48)=67,

a(49)=347, a(50)=31, a(51)=19, a(52)=17, a(53)=347,

a(54)=11, a(55)=13, a(56)=13, a(57)=31, a(58)=73,

a(59)=641, a(60)=5

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

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

For smallest mother factorial prime p of order n see A139075

For smallest father factorial prime p of order n see A139207

For smallest son factorial prime p of order n see A139206

Summary added by Robert Price, Nov 25 2010:

a(1:20)=2,2,3,5,7,3,11,7,26737,5,13,5,>60000,17,7,13,43,7,31,5

a(21:40)=7,>60000,1579,7,47,17,17,7,241,5,61,11,17,17,29,11,61,103,89,7

a(41:60)=131,11,71,13,7,43,73,67,347,31,19,17,347,11,13,13,31,73,641,5

a(61:80)=89,31,13,13,17,11,71,19,131,7,151,7,>10000,641,73,43,17,331,113,11

a(81:100)=13,67,>10000,7,1999,89,31,11,>10000,19,19,31,607,71,61,11,761,23,>10000,83

Examples

			a(1) = 2 because 2 is the first prime and 2!/1 + 1 = 3 is prime
a(2) = 2 because 2 is the first prime and 2!/2 + 1 = 2 is prime
a(3) = 3 because 3!/3 + 1 = 3 is prime
		

Crossrefs

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

Programs

Mathematica

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

Extensions

a(9)-a(12) by Robert Price, Dec 19 2010