A002981 -id:A002981 - cp's OEIS frontend

A035093 Smallest k such that k*n! + 1 is prime.

1, 1, 1, 3, 2, 3, 3, 4, 3, 3, 1, 2, 3, 13, 7, 4, 5, 2, 7, 17, 15, 18, 3, 6, 3, 16, 1, 4, 7, 20, 8, 3, 9, 5, 2, 7, 1, 3, 10, 3, 1, 29, 7, 9, 45, 8, 3, 6, 35, 66, 2, 20, 2, 4, 25, 52, 14, 34, 24, 6, 10, 22, 38, 16, 20, 91, 69, 12, 19, 20, 21, 42, 1, 5, 33, 77, 1, 2, 12, 29, 193, 74, 40, 55, 19

Offset: 1

Labos Elemer

nonn

This is one possible generalization of "the least prime problem" for n*k+1 arithmetic progression when n is replaced by n!, a special difference.

			a(7)=3 because in progression of 5040*k+1 the terms 5041 and 10081 are not prime and so 15121 is the first prime.

Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. Special case for k=1 is A002981.

Table[k = 1; While[! PrimeQ[1 + k*n!], k++]; k, {n, 85}] (* T. D. Noe, Nov 04 2013 *)

a(n) = my(k=1); while(!isprime(k*n!+1), k++); k; \\ Michel Marcus, Sep 26 2020

a(80) corrected by Alex Ratushnyak, Nov 03 2013

Simpler title by Sean A. Irvine, Sep 25 2020

A076683 Numbers k such that 7*k! + 1 is prime.

Original entry on oeis.org

3, 7, 8, 15, 19, 29, 36, 43, 51, 158, 160, 203, 432, 909, 1235, 3209, 8715, 9707

Offset: 1

Phillip L. Poplin (plpoplin(AT)bellsouth.net), Oct 25 2002

a(17) > 5830. - Jinyuan Wang, Feb 05 2020

a(19) > 12000. - Michael S. Branicky, Jul 04 2024

			k = 3 is here because 7*3! + 1 = 43 is prime.

Cf. A002981, A051915, A076679, A076680, A076681, A076682, A178488, A180626, A126896.

is(k) = ispseudoprime(7*k!+1); \\ Jinyuan Wang, Feb 05 2020

from sympy import isprime
from math import factorial
def aupto(m): return [k for k in range(m+1) if isprime(7*factorial(k)+1)]
print(aupto(300)) # Michael S. Branicky, Mar 07 2021

a(17)-a(18) from Michael S. Branicky, Jul 04 2024

A178488 Numbers k such that 8*k! + 1 is prime.

Original entry on oeis.org

2, 4, 9, 10, 11, 12, 15, 25, 31, 46, 53, 78, 318, 615, 955, 1646, 2669, 2672, 3515, 7689

Offset: 1

Robert G. Wilson v, Sep 13 2010 and M. F. Hasler, Sep 16 2010

a(20) > 3810. - Jinyuan Wang, Feb 05 2020

a(21) > 12000. - Michael S. Branicky, Jul 03 2024

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002981, A051915, A076679, A076680, A076681, A076682, A076683, A180626, A126896.

fQ[n_] := PrimeQ[8 n! + 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst

for(k=1, 999, ispseudoprime(8*k!+1) & print1(k, ", "))

ABC2 8*$a!+1
a: from 1 to 1000 // Jinyuan Wang, Feb 05 2020

a(16)-a(19) from Jinyuan Wang, Feb 05 2020

a(20) from Michael S. Branicky, Jul 02 2024

A180626 Numbers k such that 9*k! + 1 is prime.

Original entry on oeis.org

2, 6, 7, 10, 13, 15, 24, 29, 33, 44, 98, 300, 548, 942, 1099, 1176, 1632, 1794, 3676, 3768, 4804, 6499, 8049, 8164, 8917, 10270, 11610, 11959

Offset: 1

Robert G. Wilson v, Sep 13 2010

Tested to 4500. - Robert G. Wilson v, Sep 28 2010

a(22) > 5235. - Jinyuan Wang, Feb 05 2020

Index entries for sequences related to numbers of primes of the form k*n!+-1

Cf. A002981, A051915, A076679, A076680, A076681, A076682, A076683, A178488, A126896.

fQ[n_] := PrimeQ[9 n! + 1]; k = 0; lst = {}; While[k < 1501, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst

is(k) = ispseudoprime(9*k!+1); \\ Jinyuan Wang, Feb 05 2020

a(17)-a(20) from Robert G. Wilson v, Sep 28 2010
a(21) from Jinyuan Wang, Feb 05 2020
a(22) from Michael S. Branicky, May 27 2023
a(23)-a(28) from Michael S. Branicky, Jul 12 2024

A204659 Numbers n such that n!9-1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 15, 20, 23, 27, 30, 44, 51, 62, 80, 90, 95, 114, 129, 138, 150, 152, 156, 182, 201, 216, 293, 332, 342, 393, 411, 414, 419, 525, 668, 743, 800, 972, 1034, 1266, 1785, 1869, 2777, 3561, 3780, 4106, 4328, 4428, 4556, 4574, 4629, 5001, 5397, 6315

Offset: 1

M. F. Hasler, Jan 17 2012

n!9 = A114806(n).
a(74) > 50000. - Robert Price, Jun 14 2012
a(1)-a(73) are proved prime by the deterministic test of pfgw. - Robert Price, Jun 14 2012

Robert Price, Table of n, a(n) for n = 1..81
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!9-1.

Cf. A204657, A204658, A204660, A204661, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], PrimeQ[MultiFactorial[#, 9] - 1] & ] (* Robert Price, Apr 19 2019 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\9,n-9*i)-1)& print1(n","))

a(47)-a(73) from Robert Price, Jun 14 2012

Extended b-file adding a(74)-a(81) using data from Ken Davis link by Robert Price, Apr 19 2019

A204660 Numbers n such that n!9+1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 24, 25, 32, 40, 43, 48, 49, 50, 57, 60, 71, 73, 82, 83, 86, 97, 105, 114, 121, 142, 147, 159, 168, 195, 205, 210, 212, 233, 262, 288, 289, 300, 309, 316, 323, 356, 403, 447, 505, 514, 553, 735, 739, 777

Offset: 1

M. F. Hasler, Jan 17 2012

n!9 = A114806(n).
a(107) > 50000. - Robert Price, Jun 18 2012
a(1)-a(106) verified prime by deterministic test of PFGW. - Robert Price, Jun 18 2012

Robert Price, Table of n, a(n) for n = 1..106
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!9+1.

Cf. A204657, A204658, A204659, A204661, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 9] + 1] & ] (* Robert Price, Apr 19 2019 *)
Select[Range[0,800],PrimeQ[Times@@Range[#,1,-9]+1]&] (* Harvey P. Dale, Aug 19 2021 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\9,n-9*i)+1)& print1(n","))

A055487 Least m such that phi(m) = n!.

Original entry on oeis.org

1, 3, 7, 35, 143, 779, 5183, 40723, 364087, 3632617, 39916801, 479045521, 6227180929, 87178882081, 1307676655073, 20922799053799, 355687465815361, 6402373865831809, 121645101106397521, 2432902011297772771, 51090942186005065121, 1124000727844660550281, 25852016739206547966721, 620448401734814833377121, 15511210043338862873694721, 403291461126645799820077057, 10888869450418352160768000001, 304888344611714964835479763201

Offset: 1

Labos Elemer, Jun 28 2000

nonn

Erdős believed (see Guy reference) that phi(x) = n! is solvable.
Factorial primes of the form p = A002981(m)! + 1 = k! + 1 give the smallest solutions for some m (like m = 1,2,3,11) as follows: phi(p) = p-1 = A002981(m)!.
According to Tattersall, in 1950 H. Gupta showed that phi(x) = n! is always solvable. - Joseph L. Pe, Oct 01 2002
A123476(n) is a solution to the equation phi(x)=n!. - T. D. Noe, Sep 27 2006
From M. F. Hasler, Oct 04 2009: (Start)
Conjecture: Unless n!+1 is prime (i.e., n in A002981), a(n)=pq where p is the least prime > sqrt(n!) such that (p-1) | n! and q=n!/(p-1)+1 is prime.
Probably "least prime > sqrt(n!)" can also be replaced by "largest prime <= ceiling(sqrt(n!))". The case "= ceiling(...)" occurs for n=5, sqrt(120) = 10.95..., p=11, q=13.
a(n) is the first element in row n of the table A165773, which lists all solutions to phi(x)=n!. Thus a(n) = A165773((Sum_{kA055506(k)) + 1). The last element of each row (i.e., the largest solution to phi(x)=n!) is given in A165774. (End)

R. K. Guy, (1981): Unsolved problems In Number Theory, Springer - page 53.
Tattersall, J., "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, p. 162.

Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
P. Erdős and J. Lambek, Problem 4221, Amer. Math. Monthly, 55 (1948), 103.

Cf. A055486, A055488, A055489, A055506, A000010, A000142.

Cf. A123476, A165773, A165774.

Array[Block[{k = 1}, While[EulerPhi[k] != #, k++]; k] &[#!] &, 10] (* Michael De Vlieger, Jul 12 2018 *)

a(n) = Min{m : phi(m) = n!} = Min{m : A000010(m) = A000142(n)}.

More terms from Don Reble, Nov 05 2001

a(21)-a(28) from Max Alekseyev, Jul 09 2014

A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 6, 4, 16, 2, 4, 2, 24, 4, 8, 8, 8, 4, 16, 8, 4, 4, 8, 4, 4, 16, 32, 2, 8, 2, 2, 4, 8, 4, 32, 2, 16, 4, 16, 16, 128, 16, 32, 32, 4, 16, 8, 4, 32, 32, 16, 64, 64, 32, 64, 32, 4, 8, 16, 16, 32, 16, 64, 16, 128, 4, 64, 32, 32, 8, 16, 32, 128, 8

Offset: 2

Vladeta Jovovic, Sep 11 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 2..135
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002582, A027423, A002981, A002982, A064144.

Do[ Print[ DivisorSigma[0, n! - 1]], {n, 2, 40} ]
DivisorSigma[0,Range[2,80]!-1] (* Harvey P. Dale, Aug 17 2024 *)

{ f=1; for (n=2, 100, f*=n; if (n>1, a=numdiv(f - 1), a=0); write("b064145.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 09 2009

More terms from Robert G. Wilson v, Oct 04 2001
a(51)-a(76) from Harry J. Smith, Sep 09 2009
Ambiguous term a(1) removed by Max Alekseyev, May 06 2022

A076682 Numbers k such that 6*k! + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 12, 13, 18, 24, 38, 48, 60, 113, 196, 210, 391, 681, 739, 778, 1653, 1778, 1796, 1820, 2391, 2505, 4595, 8937

Offset: 1

Phillip L. Poplin (plpoplin(AT)bellsouth.net), Oct 25 2002

a(29) > 5800. - Jinyuan Wang, Feb 05 2020

a(30) > 12000. - Michael S. Branicky, Jul 04 2024

			k = 3 is here because 6*3! + 1 = 37 is prime.

Cf. A002981, A051915, A076679, A076680, A076681, A076683, A178488, A180626, A126896.

is(k) = ispseudoprime(6*k!+1); \\ Jinyuan Wang, Feb 05 2020

a(26) inserted by and a(29) from Michael S. Branicky, Jul 03 2024

A125162 a(n) is the number of primes of the form k! + n, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 0, 1, 1, 5, 1, 3, 0, 1, 1, 6, 1, 7, 0, 1, 1, 6, 0, 1, 0, 1, 1, 6, 1, 9, 0, 0, 0, 3, 1, 11, 0, 1, 1, 9, 1, 5, 0, 1, 1, 10, 0, 2, 0, 1, 1, 9, 0, 2, 0, 1, 1, 10, 1, 9, 0, 0, 0, 3, 1, 9, 0, 1, 1, 8, 1, 9, 0, 0, 0, 5, 1, 9, 0, 1, 1, 11, 0, 1, 0, 1, 1, 8, 0, 3, 0, 0, 0, 2, 1, 10, 0, 1, 1, 10, 1

Offset: 1

Alexander Adamchuk, Nov 21 2006

nonn

Note the triples of consecutive zeros in a(n) for n = {{32,33,34}, {62,63,64}, {74,75,76}, {92,93,94}, {116,117,118}, {122,123,124}, {140,141,142}, {152,153,154}, {158,159,160}, {182,183,184}, {200,201,202}, {206,207,208}, {212,213,214}, {218,219,220}, {242,243,244}, {272,273,274}, {284,285,286}, ...}. The middle index of most zero triples is a multiple of 3. See A125164.
The first consecutive quintuple of zeros has indices n = {294,295,296,297,298}, where the odd zero index n = 295 is not a multiple of 3.
Also for n >= 2, a(n) is the number of primes of the form k! + n for all k, since n divides k! + n for k >= n. Note that it is not known whether there are infinitely many primes of the form k! + 1; see A088332 for such primes and A002981 for the indices k. - Jianing Song, Jul 28 2018

			a(n) is the length of n-th row in the table of numbers k such that k! + n is a prime, 1 <= k <= n.
   n:  numbers k
   -------------
   1:  {1},
   2:  {1},
   3:  {2},
   4:  {1},
   5:  {2, 3, 4},
Thus a(1)-a(4) = 1, a(5) = 3.
See Example table link for more rows.

Michel Marcus, Example table

Cf. A125163 (indices of 0), A125164 (triples).

Table[Length[Select[Range[n],PrimeQ[ #!+n]&]],{n,1,300}]

a(n)=c=0;for(k=1,n,if(ispseudoprime(k!+n),c++));c
vector(100,n,a(n)) \\ Derek Orr, Oct 15 2014

Name clarified by Jianing Song, Jul 28 2018

Edited by Michel Marcus, Jul 29 2018

cp's OEIS Frontend

A035093 Smallest k such that k*n! + 1 is prime.

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A076683 Numbers k such that 7*k! + 1 is prime.

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A178488 Numbers k such that 8*k! + 1 is prime.

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PFGW

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A180626 Numbers k such that 9*k! + 1 is prime.

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A204659 Numbers n such that n!9-1 is prime.

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A204660 Numbers n such that n!9+1 is prime.

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A055487 Least m such that phi(m) = n!.

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