A037083 -id:A037083 - cp's OEIS frontend

A204663 Numbers n such that n!8 + 2 is prime.

0, 1, 3, 5, 9, 13, 15, 21, 23, 27, 33, 35, 45, 53, 55, 57, 75, 79, 109, 197, 221, 227, 267, 333, 413, 545, 695, 703, 801, 967, 1029, 1329, 1351, 1475, 1549, 1757, 2173, 2861, 3161, 3167, 3885, 4681, 4965, 6277, 6655, 8477, 9821, 9959, 10269, 17999, 23349, 29347, 29477, 30181, 34133, 36687, 40985, 43395, 47499

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).
See also links in A156165.
For odd k, n!k +-2 is even for all n > k and thus cannot be prime.
a(60) > 50000. - Robert Price, Aug 19 2012

C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75.
Ken Davis, Status of Search for Multifactorial Primes.

Cf. A204657, A204658, A204659, A204660, A204661, A204662, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

Select[Range[0,9999], PrimeQ[Product[# - 8i,{i, 0, Floor[(# - 2)/8]}] + 2] &] (* Indranil Ghosh, Mar 13 2017 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)+2)& print1(n","))

a(39)-a(59) from Robert Price, Aug 19 2012

A204664 Numbers n such that n!8-2 is prime.

Original entry on oeis.org

4, 5, 7, 9, 11, 15, 17, 25, 27, 33, 47, 59, 63, 77, 87, 89, 93, 95, 107, 119, 127, 133, 139, 193, 201, 217, 269, 291, 369, 373, 435, 445, 669, 803, 831, 859, 907, 1271, 1705, 1743, 1849, 3087, 3189, 3497, 4221, 4475, 5119, 6013, 8023, 9237, 12755, 16501, 16747, 17021, 17309, 20671, 21539, 28377, 33625, 35645, 36831, 54663, 56223, 65299, 66159, 68121, 69339, 70579, 73511, 77745, 94601

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).
See also links in A156165.
For odd k, n!k +- 2 is even for all n > k and thus cannot be prime.
a(62) > 50000. - Robert Price, Aug 27 2012
The first 10 associated primes: 2, 3, 5, 7, 31, 103, 151, 3823, 16927, 126223. - Robert Price, Mar 10 2017
a(72) > 10^5. - Robert Price, Apr 24 2017

C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75
Ken Davis, Status of Search for Multifactorial Primes.

Cf. A204657, A204658, A204659, A204660, A204661, A204662, A204663, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[4, 50000], PrimeQ[MultiFactorial[#, 8] - 2] &] (* Robert Price, Mar 10 2017 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)-2)& print1(n","))

a(46)-a(61) from Robert Price, Aug 27 2012

a(62)-a(71) from Robert Price, Apr 24 2017

A135726 Primes of the form k!!! - 1 = A007661(k) - 1, k > 0.

Original entry on oeis.org

2, 3, 17, 79, 4188799, 2504902399, 254561089305599, 13106744139423334399999, 8483004771271882804592639999, 706526001186582385898210420541078864497278132689882316799999999, 353401447088718405944982176443380974931403135679741865504466985287679999999999

Offset: 1

M. F. Hasler, Nov 26 2007

nonn

Sequence A084438 gives the easier-to-read n-values.

All terms greater than a(3) seem to end in the digit 9, or many 9 digits. a(17) ends with 51 9 digits. - Harvey P. Dale, Nov 28 2019

			a(4) = 79 = 8*5*2 - 1 = 8!!! - 1 is the 4th prime of that form.

Harvey P. Dale, Table of n, a(n) for n = 1..17
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!3-1.
ProthSearch.net (on web.archive.org), n3minus.txt.

Cf. A007661, A037082, A037083, A084438.

Select[Table[Times@@Range[n,1,-3],{n,150}]-1,PrimeQ] (* Harvey P. Dale, Nov 28 2019 *)

A007661(n) = prod(i=1,(n-1)\3,n-=3,n+!n)
for(n=1,999,if(isprime(A007661(n)-1),print1(A007661(n)-1,",")))

a(n) = A007661(A084438(n)) - 1. - Elmo R. Oliveira, Feb 25 2025

A100013 Number of prime factors in n!+7 (counted with multiplicity).

Original entry on oeis.org

3, 3, 2, 1, 1, 1, 1, 3, 3, 3, 3, 2, 3, 3, 4, 2, 2, 3, 3, 5, 5, 5, 3, 4, 3, 2, 4, 5, 5, 4, 7, 6, 4, 4, 7, 2, 5, 4, 7, 4, 5, 3, 4, 6, 5, 4, 3, 3, 5, 6, 3, 5, 6, 3, 3, 7, 4, 5, 5, 2, 4, 4, 5, 4, 2, 4, 3, 5, 2, 5, 7, 4, 7, 5, 5, 3, 5, 4, 6, 6, 8, 5

Offset: 0

Jonathan Vos Post, Nov 18 2004

			Example 1!+7 = 2^3 so a(1) = 3.
a(3) = a(4) = a(5) = a(6) = 1 because 3!+1 = 13, 4!+7 = 31, 5!+1 = 127, 6!+7 = 727 and these are all primes. a(11) = a(15) = a(16) = a(25) = a(35) = a(59) = 2 because 11!+7 = 39916807 = 7 * 5702401, 15!+7 = 1307674368007 = 7 * 186810624001, 16!+7 = 20922789888007 = 7 * 2988969984001, 25!+7 = 15511210043330985984000007 = 7 * 2215887149047283712000001, 35!+7 = 10333147966386144929666651337523200000007 = 7 *
1476163995198020704238093048217600000001 and 59!+7 = 138683118545689835737939019720389406345902876772687432540821294940160000000000007 = 7 * 19811874077955690819705574245769915192271839538955347505831613562880000000000001 are all semiprimes.

C. Caldwell and H. Dubner, "Primorial, factorial and multifactorial primes," Math. Spectrum, 26:1 (1993/4) 1-7.

Chris Caldwell, multifactorial prime (another Prime Pages' Glossary entries).
Ken Davis, Status of Search for Multifactorial Primes.
Sean A. Irvine, Factorizations of n!+7

Cf. A002981, A037083, A037083, A085150, A085148, A085146, A062701, A055487, A062702.

More terms from Sean A. Irvine, Sep 20 2012

A271392 Integers k such that 3*k!!! + 1 is prime where k!!! is A007661(k).

Original entry on oeis.org

2, 4, 5, 8, 9, 15, 16, 23, 27, 32, 34, 35, 38, 40, 46, 54, 57, 83, 87, 97, 162, 165, 223, 235, 282, 488, 503, 575, 673, 823, 857, 885, 965, 1112, 1401, 2288, 2569, 2788, 3133, 3539, 4070, 4654, 5020, 5613, 6720, 7773, 11256, 18023, 22196

Offset: 1

Altug Alkan, Apr 06 2016

Corresponding primes are 7, 13, 31, 241, 487, 87481, 174721, 289027201, 21427701121, ...

			4 is a term because 3*4!!! + 1 = 13 is prime.

Cf. A007661, A037083, A084438, A217647, A215779, A271396.

is(k) = ispseudoprime(3*prod(i=0, (k-2)\3, k-3*i) + 1); \\ Jinyuan Wang, Jun 09 2021

a(47) from Jinyuan Wang, Jun 09 2021

a(48)-a(49) from Michael S. Branicky, Aug 10 2024

A271396 Integers k such that 3*k!!! - 1 is prime where k!!! is A007661(k).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 10, 17, 25, 28, 31, 37, 38, 39, 46, 47, 49, 55, 67, 82, 85, 94, 98, 115, 120, 129, 167, 214, 216, 267, 293, 580, 732, 857, 993, 1012, 1069, 1308, 1430, 2366, 2974, 4017, 4870, 9034, 9061, 9752, 10657, 13847, 25390

Offset: 1

Altug Alkan, Apr 06 2016

Corresponding primes are 2, 2, 5, 11, 29, 53, 83, 239, 839, 628319, 1825823999, 51123071999, ...

			4 is a term because 3*4!!! - 1 = 11 is prime.

Cf. A007661, A037083, A084438, A217647, A215779, A271392.

Select[Range[0,5000],PrimeQ[3Times@@Range[#,1,-3]-1]&] (* The program generates the first 45 terms of the sequence. *) (* Harvey P. Dale, Mar 29 2025 *)

is(k) = ispseudoprime(3*prod(i=0, (k-2)\3, k-3*i) - 1); \\ Jinyuan Wang, Jun 09 2021

a(46)-a(50) from Jinyuan Wang, Jun 09 2021

a(51) from Michael S. Branicky, Aug 09 2024

A110094 Startorial primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 719, 5039, 1451521, 2903041, 5806081, 46448639, 92897281, 371589121, 10032906239, 30098718719, 270888468479, 812665405439, 7313988648961, 21941965946881, 89874292518420479

Offset: 1

Jonathan Vos Post, Sep 04 2005

These are primes of the form A109834 startorials (base 10) +1 or -1. This is by analogy to factorial primes (A002981), superfactorial primes (A073828), hyperfactorial primes, ultrafactorial primes (comment in A046882), subfactorial primes (A100015), double factorial primes (A080778), multifactorial primes (A037083).

Cf. A000040, A002981, A073828, A080778, A037083, A100015, A103317, A046882, A109834.

{a(n)} = {A109834(k)+1 an element of A000040, or A109834(k)-1 an element of A000040, for some k}.

A274386 Triple factorial primes: primes which are within 1 of a triple factorial number.

Original entry on oeis.org

2, 3, 5, 11, 17, 19, 29, 79, 163, 281, 881, 209441, 4188799, 264539521, 2504902399, 72642169601, 254561089305599, 9927882482918401, 26582634158080001, 13106744139423334399999, 141383412854531380076544001, 427380210218181008588800001

Offset: 1

Arkadiusz Wesolowski, Jun 19 2016

nonn

Union of A037082 and A135726.

			a(7) = 29 = 4*7 + 1 = 7!!! + 1 is the 7th prime of that form.
a(8) = 79 = 2*5*8 - 1 = 8!!! - 1 is the 8th prime of that form.

Index entries for sequences related to factorial numbers

Cf. A007661, A037082, A037083, A084438, A135726.

r:=59; I:=[1, 1, 2]; lst1:=[n le 3 select I[n] else (n-1)*Self(n-3): n in [1..r]]; lst2:=[]; for c in [1..r] do a:=lst1[c]; for s in [-1..1 by 2] do p:=a+s; if IsPrime(p) and not p in lst2 then Append(~lst2, p); end if; end for; end for; lst2;

Select[Union@ Flatten@ Map[{# - 1, # + 1} &, Table[With[{q = Quotient[n + 2, 3]}, 3^q q! Binomial[n/3, q]], {n, 0, 58}]], PrimeQ] (* Michael De Vlieger, Jun 21 2016, after Jan Mangaldan at A007661 *)
Select[Union[Flatten[#+{1,-1}&/@Table[Times@@Range[n,1,-3],{n,100}]]],PrimeQ] (* Harvey P. Dale, Sep 05 2022 *)

cp's OEIS Frontend

A204663 Numbers n such that n!8 + 2 is prime.

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A204664 Numbers n such that n!8-2 is prime.

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A135726 Primes of the form k!!! - 1 = A007661(k) - 1, k > 0.

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A100013 Number of prime factors in n!+7 (counted with multiplicity).

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A271392 Integers k such that 3*k!!! + 1 is prime where k!!! is A007661(k).

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Programs

PARI

Extensions

A271396 Integers k such that 3*k!!! - 1 is prime where k!!! is A007661(k).

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Comments

Examples

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Programs

Mathematica

PARI

Extensions

A110094 Startorial primes.

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Comments

Crossrefs

Formula

A274386 Triple factorial primes: primes which are within 1 of a triple factorial number.

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