A136020 -id:A136020 - cp's OEIS frontend

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

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

A139207 Smallest father factorial prime p of order n = smallest prime of the form (p!-n)/n where p is prime.

Original entry on oeis.org

5, 2, 2947253997913233984847871999999, 29, 23, 19, 719, 4989599, 39520825343999, 11, 11058645491711999, 419, 479001599, 359, 7, 860234568201646565394748723848806399999999

Offset: 1

Artur Jasinski, Apr 11 2008

For smallest daughter factorial prime p of order n (smallest p such that (p!+n)/n = p!/n + 1 is prime) see A139074.
For smallest son factorial prime p of order n = smallest prime of the form (p!-n)/n where p is prime see A139206.
For more terms see A139206.

Cf. A139074, A139189, A139190, A139191, A139192, A139193, A139194, A139195, A139196, A139197, A139198, A136019, A136020, A136026, A136027.

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

A139073 Smallest prime number of the form (n+k!)/n.

Original entry on oeis.org

2, 2, 3, 7, 1009, 2, 5702401, 631, 4481, 13, 566092801, 3, 23452949585516450807808000001, 259201, 337, 2521, 3553839003727872684550301886383176323956736000000001, 41

Offset: 1

Artur Jasinski, Apr 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..58

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

a = {}; Do[ k = 1; While[ ! PrimeQ[ (k! + n)/n ], k++ ]; AppendTo[ a, (k! + n)/n ], {n, 1, 100} ]; a [Corrected May 06 2008]

a(n)=my(k,t);until(denominator(t=k++!/n+1)==1&&ispseudoprime(t),);t \\ Charles R Greathouse IV, Jul 19 2011

a(n) = (n + A139072(n)!)/n. - Amiram Eldar, Oct 14 2024

A139091 a(n) = largest prime divisor of the number prime(n)!/9 + 1.

Original entry on oeis.org

17, 827, 22319071, 1718296754087, 35662591735219, 477262171871, 1609727002420791262479701, 146215297537890243023, 2020914387433686758547638152441, 1073774770807266077323

Offset: 4

Artur Jasinski, Apr 08 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 4..26

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

a = {}; Do[w = FactorInteger[(Prime[n]! + 9)/9]; AppendTo[a, Last[w][[1]]], {n, 4, 16}]; a

A139090 a(n) = smallest prime divisor of the number prime(n)!/9 + 1.

Original entry on oeis.org

3, 31, 31, 23, 379, 83, 610301, 293, 101, 47, 281, 127, 278174297, 2971, 109, 5090615254324820333, 46411, 106087, 269, 288931, 59047158151, 120871, 373, 19140822523, 56595118147, 1708207, 331, 38749, 157, 2927, 2143

Offset: 4

Artur Jasinski, Apr 08 2008

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

a = {}; Do[w = FactorInteger[(Prime[n]! + 9)/9]; AppendTo[a, First[w][[1]]], {n, 4, 16}]; a
Table[FactorInteger[p!/9+1][[1,1]],{p,Prime[Range[4,35]]}] (* Harvey P. Dale, Sep 19 2020 *)

More terms from Jon E. Schoenfield, Jul 16 2010

A348153 Primes for which there is no pair (k,q) with k a positive integer and q another prime, such that p=q*(2k+1)-2k.

Original entry on oeis.org

2, 3, 5, 17, 41, 73, 89, 97, 137, 193, 233, 257, 313, 353, 449, 457, 569, 641, 809, 857, 929, 1033, 1049, 1097, 1129, 1153, 1193, 1217, 1289, 1409, 1433, 1601, 1609, 1697, 1753, 1889, 1913, 1993, 2081, 2137, 2153, 2273, 2297, 2393, 2473, 2617, 2633, 2657, 2689, 2713, 2753, 2777, 2969

Offset: 1

René Gy, Oct 03 2021

nonn

There are primes p for which there exist a positive integer k and another prime q such that p=q*(2k+1)-2k. See A136020, A091180, A136061 and the subsequent sequences. Such k is called an "order" of the prime p. Note that q is necessarily larger than 2 and that 4*k is necessarily smaller than p-1. A prime may belong to more than one order, but the primes listed in the present sequence do not belong to any order.

As soon as they are larger than 8, all members minus 1 are multiples of 8.

Cf. A136020, A091180, A136061.

lim = 2000; p = 2; listc = {}; listp = {}; While[p < lim, n = 1;
While[n <= (p - 3)/4,
  If[PrimeQ[(p + 2 n)/(2 n + 1)], n = 2*p, n = n + 1]];
If[n == 2*p, AppendTo[listc, p]]; AppendTo[listp, p];
p = NextPrime[p]]; Complement[listp, listc]

isok(p) = {if (isprime(p), for (k=1, (p-3)/4, my(q = (p+2*k)/(2*k+1)); if ((denominator(q)==1) && isprime(q), return(0));); return (1););} \\ Michel Marcus, Oct 07 2021

More terms from Michel Marcus, Oct 04 2021

cp's OEIS Frontend

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

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

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A139207 Smallest father factorial prime p of order n = smallest prime of the form (p!-n)/n where p is prime.

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A139073 Smallest prime number of the form (n+k!)/n.

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A139091 a(n) = largest prime divisor of the number prime(n)!/9 + 1.

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A139090 a(n) = smallest prime divisor of the number prime(n)!/9 + 1.

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A348153 Primes for which there is no pair (k,q) with k a positive integer and q another prime, such that p=q*(2k+1)-2k.

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Keywords

Comments

Crossrefs

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Mathematica

PARI

Extensions