A136027 -id:A136027 - cp's OEIS frontend

A136075 Father primes of order 6.

103, 181, 233, 311, 389, 571, 701, 883, 1039, 1091, 1429, 1481, 1663, 1949, 2053, 2131, 2339, 2521, 2963, 3041, 3119, 3613, 3691, 3821, 4003, 4133, 4523, 4549, 4679, 4783, 4861, 5641, 5849, 5953, 6343, 6551, 7253, 7331, 7643, 8059, 8423, 8501, 8761

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136076, A136077, A136078, A136079, A136080.

n = 6; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136076 Father primes of order 7.

Original entry on oeis.org

59, 89, 179, 269, 359, 449, 479, 569, 659, 719, 809, 929, 1019, 1109, 1259, 1559, 1619, 1709, 1979, 2069, 2099, 2459, 2609, 2699, 2729, 2879, 2909, 2969, 2999, 3359, 3449, 3779, 4049, 4079, 4229, 4259, 4409, 4679, 5309, 5399, 5519, 5849, 6029, 6299, 6329

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136077, A136078, A136079, A136080.

n = 7; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136077 Father primes of order 8.

Original entry on oeis.org

67, 101, 509, 1019, 1223, 1427, 1733, 2243, 2549, 2957, 4079, 4283, 4793, 5303, 6833, 7547, 7649, 7853, 8363, 8669, 9587, 9689, 11117, 11933, 12239, 12647, 12953, 15809, 16217, 18869, 19583, 20297, 20807, 21419, 21929, 22133, 23357, 24683, 25703

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136075.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136078, A136079, A136080.

n = 8; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136078 Father primes of order 9.

Original entry on oeis.org

113, 151, 227, 379, 569, 607, 797, 911, 1291, 1367, 1709, 1861, 2089, 2621, 2659, 2887, 3001, 3191, 3457, 3761, 4027, 4597, 4787, 5167, 5281, 5851, 5927, 6421, 6991, 7219, 7561, 7789, 8017, 9689, 10601, 10867, 11171, 11399, 11437, 11551, 11779

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136076. For father primes of order 8 see A136077.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136079, A136080.

n = 9; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a
Select[Prime[Range[1500]],PrimeQ[(#-18)/19]&] (* Harvey P. Dale, Mar 26 2023 *)

A136079 Father primes of order 10.

Original entry on oeis.org

83, 167, 251, 293, 419, 503, 797, 881, 1259, 1301, 1427, 1511, 1553, 1889, 2141, 2267, 2309, 2393, 2687, 2897, 2939, 3191, 3527, 3779, 3821, 4073, 4157, 4451, 4703, 4787, 5039, 5081, 5417, 5669, 5711, 6173, 6551, 6971, 7307, 7349, 7433, 7559, 7727, 7853

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n, see A136026 (also definition). For father primes of orders 1,2,...,9, see A094524, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, respectively.
From Bob Selcoe, Apr 25 2014: (Start)
In general, a father prime, p', of order k is of the form p'=2k+(2k+1)*p for some prime, p. In this sequence, k=10, and so each prime is of the form p'=20+21p where p ranges over {3,7,11,13,19,23,...}. Thus a father prime p' has order k when (p'-2k)/(2k+1) is prime.
Father primes (p') of order k will be of the form: p'(mod (4k+2))=4k+1, or p'=(4k+2)*j-1, j>=2. For this sequence: k=10, 4k+2=42; j={2,4,6,7,10,12,...}. So for example, j=7 generates a father prime because 42*7-1 = 293 AND (293-(2*10))/(2*10+1) = 13, since both 13 and 293 are prime. Note that not all j such that (4k+2)*j-1 is prime will produce a father prime. In this example, when j=11, 42*11-1=461 (prime); but (461-(2*10))/(2*10+1) = 21 (not prime). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, A136080.

n = 10; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136080 Father primes of order 11.

Original entry on oeis.org

137, 1103, 1931, 2069, 2621, 3449, 3863, 4001, 4139, 5381, 5519, 6761, 8831, 8969, 10211, 13109, 13523, 13799, 15731, 18353, 19319, 21803, 23321, 23459, 25253, 25391, 28151, 28289, 28979, 29531, 29669, 31601, 32429, 32843, 33119, 34361

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest father primes of order n see A136026 (also definition). For father primes of order 1 see A094524. For father primes of order 2 see A136071. For father primes of order 3 see A136072. For father primes of order 4 see A136073. For father primes of order 5 see A136074. For father primes of order 6 see A136075. For father primes of order 7 see A136076. For father primes of order 8 see A136077. For father primes of order 9 see A136078. For father primes of order 10 see A136079.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023208, A094524, A136019, A136020, A136026, A136027, A136071, A136072, A136073, A136074, A136075, A136076, A136077, A136078, A136079.

n = 11; a = {}; Do[If[PrimeQ[(Prime[k] - 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A139072 Smallest parameter k such that (n+k!)/n is prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 3, 11, 7, 8, 5, 13, 4, 28, 10, 7, 8, 43, 6, 21, 5, 7, 16, 48, 4, 14, 17, 9, 7, 241, 5, 61, 11, 17, 17, 8, 10, 44, 38, 16, 6, 131, 9, 63, 12, 6, 43, 73, 9, 15, 10, 19, 14, 64, 11, 12, 9, 24, 32, 641, 5, 89, 31, 8, 8, 14, 11, 71, 19, 25, 7, 151, 6, 78, 62, 15, 35, 15, 22, 87

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

a(n) >= A002034(n). - Charles R Greathouse IV, Jul 15 2011

a(878) > 5000. - Jinyuan Wang, Apr 01 2020

Jinyuan Wang, Table of n, a(n) for n = 1..877

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

a = {}; Do[k = 1; While[ ! PrimeQ[(k! + n)/n], k++ ]; AppendTo[a, k], {n, 1, 100}]; a

pr(n)=denominator(n)==1 && ispseudoprime(n)
a(n)=my(k);until(pr(k++!/n+1),);k \\ Charles R Greathouse IV, Jul 15 2011

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

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Apr 08 2008, Apr 21 2008

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

			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

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

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

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

A139206 Smallest son factorial prime p of order n: smallest p such that p!/n-1 is prime.

Original entry on oeis.org

3, 3, 29, 5, 5, 5, 7, 11, 17, 5, 19, 7, 13, 7, 5, 37, 139, 19

Offset: 1

Artur Jasinski, Apr 11 2008, Apr 24 2008

For smallest daughter factorial prime p of order n (smallest p such that (p!+n)/n = p!/n + 1 is prime), see A139074.
a(19) is currently unknown, a(20)=5, a(21)=7, a(22)=19.
a(19)>10000, a(23)=71, a(24)=3361. [From Andrew V. Sutherland, Apr 23 2008]
a(25)=17, a(26)=223, a(27)=157, a(28)=7, a(29)=41, a(30)=5, a(31)=31, a(32)=71, a(33)=13, a(34)=37, a(35)=19, a(36)=7, a(37)=47, a(38)=53, a(39)=13, a(40)=5, a(41)=127, a(42)=13, a(43)=67, a(44)=11, a(45)=17, a(46)=43, a(47)=71, a(48)=11, a(49)=19, a(50)=29, a(51)=17, a(52)=17, a(53)>10000.
a(19)>25000, a(53)>25000. [From Sean A. Irvine, Nov 14 2010]
a(54)=11, a(55)=23, a(56)=7, a(57)=433.
a(58)=283, a(59)>1500, a(60..66)=(7,139,239,7,11,13,13), a(67), a(68) > 1300, a(69..72)=(29,7,83,13), a(73)>1000. [From M. F. Hasler, Nov 03 2013]
Sequence A151900 (tentatively?) lists "singular indices", i.e., those for which a(n) is difficult to find. - M. F. Hasler, Nov 03 2013

Cf. A139074, A139189-A139198, A136019, A136020, A136026, A136027.

a = {}; Do[k = 1; While[ ! PrimeQ[(Prime[k]! - n)/n], k++ ]; Print[a]; AppendTo[a, Prime[k]], {n, 1, 100}]; a (*Artur Jasinski*)

a(n)=forprime(p=1,,p!%n==0 && ispseudoprime(p!/n-1) && return(p)) \\ - M. F. Hasler, Nov 03 2013

Edited by M. F. Hasler, Nov 03 2013

A139092 a(n) = number of distinct prime divisors of (9+prime(n)!)/9.

Original entry on oeis.org

3, 3, 2, 2, 2, 3, 2, 3, 4, 4, 5, 3, 3, 6, 5, 2, 3, 4, 3, 3, 4, 4, 4, 3, 7, 3, 3

Offset: 4

Artur Jasinski, Apr 08 2008

Conjecture: all prime divisors in A139089 are distinct

a(31) >= 4. - Amiram Eldar, Feb 13 2020

FactorDB, Status of a(31) = (9+127!)/9.

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A020458, A139068, A137390, A139070-A139075, A136019, A136020, A136026, A136027, A139089-A139092.

a = {}; Do[w = (Prime[n]! + 9)/9; AppendTo[a, w], {n, 4, 16}]; a
PrimeNu[(9+Prime[Range[4,25]]!)/9] (* Harvey P. Dale, Jul 25 2019 *)

a(n) = A001221(A139089(n)). - Amiram Eldar, Feb 13 2020

More terms from Jon E. Schoenfield, Jul 16 2010

a(23)-a(30) using factordb.com from Amiram Eldar, Feb 13 2020

cp's OEIS Frontend

A136075 Father primes of order 6.

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A136076 Father primes of order 7.

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A136077 Father primes of order 8.

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A136078 Father primes of order 9.

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A136079 Father primes of order 10.

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A136080 Father primes of order 11.

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A139072 Smallest parameter k such that (n+k!)/n is prime.

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

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A139206 Smallest son factorial prime p of order n: smallest p such that p!/n-1 is prime.

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