A037082 -id:A037082 - cp's OEIS frontend

A084438 Positive integers k such that k!!! - 1 = A007661(k) - 1 is prime.

3, 4, 6, 8, 20, 26, 36, 50, 60, 114, 135, 138, 248, 315, 351, 429, 642, 5505, 8793, 12086, 13580, 23109, 34626, 34706, 56282, 57675, 58298

Offset: 1

Hugo Pfoertner, Jun 25 2003

The search for multifactorial primes started by Ray Ballinger is now continued by a team of volunteers on the website of Ken Davis (see link).

			a(4) = 8 since 8!!! - 1 = 8*5*2 - 1 = 79 is the 4th prime of that form.
26!!! - 1 = 2504902399 is prime.

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!3-1.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
ProthSearch.net (on web.archive.org), n3minus.txt.

Cf. A007661, A135726, A037082, A037083.

multiFactorial[n_, k_] := If[n < 1, 1, n * multiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[multiFactorial[#, 3] - 1] & ] (* Robert Price, Apr 19 2019 *)
Select[Range[650], PrimeQ[Times @@ Range[#, 1, -3] - 1] &] (* The program generates the first 17 terms of the sequence. To generate more, change the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, May 22 2021 *)

A007661(n) = prod(i=1,(n-1)\3,n-=3,n+!n)
for(n=1,999,if(isprime(A007661(n)-1),print1(n","))) \\ M. F. Hasler, Nov 26 2007

Missing 26 inserted by M. F. Hasler, Nov 26 2007
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Edited by N. J. A. Sloane, Feb 11 2009 at the suggestion of M. F. Hasler

A242994 Numbers n such that n!3 - 3 is prime, where n!3 = n!!! is a triple factorial number (A007661).

Original entry on oeis.org

5, 10, 11, 13, 16, 22, 28, 71, 74, 94, 119, 121, 134, 157, 200, 262, 286, 484, 1039, 1045, 1190, 1595, 1679, 1772, 1789, 2410, 2920, 5039, 7919, 10462, 11846, 23293, 26705, 30781, 43694

Offset: 1

Robert Price, Aug 17 2014

Large terms correspond to probable primes. - Jens Kruse Andersen, Aug 19 2014

a(36) > 50000. - Robert Price, Oct 12 2014

			11!3-3 = 11*8*5*2-3 = 877 is prime, so 11 is in the sequence. - _Jens Kruse Andersen_, Aug 19 2014

OpenPFGW Project, Primality Tester
Henri & Renaud Lifchitz, PRP Records. Search for n!3-3

Cf. A007661, A037082, A084438, A243078.

MultiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*MultiFactorial[n - k, k]]];
lst={};Do[If[PrimeQ[MultiFactorial[n, 3] - 3], AppendTo[lst, n]], {n, 100}];lst

Links and crossrefs fixed by Jens Kruse Andersen, Aug 19 2014

a(35) from Robert Price, Oct 12 2014

A001281 Image of n under the map n->n/2 if n even, n->3n-1 if n odd.

Original entry on oeis.org

0, 2, 1, 8, 2, 14, 3, 20, 4, 26, 5, 32, 6, 38, 7, 44, 8, 50, 9, 56, 10, 62, 11, 68, 12, 74, 13, 80, 14, 86, 15, 92, 16, 98, 17, 104, 18, 110, 19, 116, 20, 122, 21, 128, 22, 134, 23, 140, 24, 146, 25, 152, 26, 158, 27, 164, 28, 170, 29, 176, 30, 182, 31, 188

Offset: 0

N. J. A. Sloane

On the set of positive integers, the orbit of any number seems to end in the orbit of 1, of 5 or of 17. Writing n=1+q*2^p with q odd, it is easily seen that for p=0,1 and p>3, some iterations of the map lead to a strictly smaller number (for n>17). The cases p=2 and p=3 may give rise to bigger loops (depending on the form of q). See sequences A135727-A135729 for maxima of the orbits and corresponding record indices. - M. F. Hasler, Nov 29 2007

R. K. Guy, Unsolved Problems in Number Theory, E16.

T. D. Noe, Table of n, a(n) for n = 0..1000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Cf. A037082.

Cf. A037084, A039500-A039505, A135727-A135730. See also A006370, A006577 (Collatz 3x+1 problem).

f := n-> if n mod 2 = 0 then n/2 else 3*n-1; fi;

Table[If[OddQ[n], 3*n-1, n/2], {n, 0, 100}] (* T. D. Noe, Jun 27 2012 *)

A001281(n)=if(n%2,3*n-1,n>>1) \\ M. F. Hasler, Nov 29 2007

f(n) = (7n-2-(5n-2)*cos(Pi*n))/4. - Robert W. Craigen (craigen(AT)fresno.edu)

G.f.: x*(2 + x + 4*x^2)/((1 - x)^2*(1 + x)^2). - Ilya Gutkovskiy, Aug 17 2016

A037083 Numbers k such that k!!! + 1 is prime (0 is included by convention).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 17, 24, 29, 39, 40, 57, 58, 59, 91, 155, 175, 245, 359, 372, 597, 864, 977, 1077, 1327, 2076, 4798, 4975, 10830, 13453, 15472, 15948, 16116, 16681, 18037, 21725, 22326, 24753, 28565, 32659, 46487, 50649, 51393, 80069, 95493

Offset: 1

G. L. Honaker, Jr.

nonn

n!!! = n*(n-3)*(n-6)*(n-9)*...

The search for multifactorial primes started by Ray Ballinger is now continued by a team of volunteers on the website of Ken Davis (see link).

Ray Ballinger, Status of Search for Multifactorial Primes
Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!3+1.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Index entries for sequences related to factorial numbers

Cf. A007661 (triple factorials), A084438, A037082.

More terms from Steven Harvey
Corrected and extended by Hugo Pfoertner, Jun 25 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
Edited by N. J. A. Sloane, Jan 14 2008

A243078 Numbers k such that k!3 - 3^2 is prime, where k!3 = k!!! is a triple factorial number (A007661).

Original entry on oeis.org

7, 8, 10, 13, 16, 17, 20, 23, 28, 29, 32, 43, 46, 47, 53, 56, 59, 61, 76, 95, 107, 139, 148, 218, 349, 764, 1009, 1130, 1183, 1429, 1516, 2072, 2471, 4937, 10204, 13993, 16249, 18166, 25733, 29033, 40090

Offset: 1

Robert Price, May 30 2014

a(42) > 50000.
k=2 and k=4 produce values (-7 and -5) whose absolute value is a prime.
Terms > 2000 correspond to probable primes.

			17!3 - 3^2 = 17*14*11*8*5*2 - 9 = 209431 is prime, so 17 is in the sequence. - _Jens Kruse Andersen_, Aug 20 2014

Henri & Renaud Lifchitz, PRP Records. Search for n!3-9.
OpenPFGW Project, Primality Tester

Cf. A007661, A037082, A084438, A123910, A242994.

MultiFactorial[n_,k_]:=If[n<1,1,If[n

a(41) from Robert Price, Sep 19 2014

A261145 Numbers n such that n!3 + 3^10 is prime, where n!3 = n!!! is a triple factorial number (A007661).

Original entry on oeis.org

2, 4, 7, 11, 25, 38, 47, 94, 95, 155, 275, 277, 292, 299, 395, 409, 614, 1409, 1963, 3422, 5243, 5884, 5971, 8527, 10882, 13223, 16406, 20851, 28886

Offset: 1

Robert Price, Nov 18 2015

Corresponding primes are: 59051, 59053, 59077, 59929, 608667049, 3091650738235049, 262134882788466747049, ...
a(30) > 50000.
Terms > 47 correspond to probable primes.

			11!3 + 3^10 = 11*8*5*2 + 59049 = 59929 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!3+59049.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A007661, A037082, A084438, A123910, A242994.

MultiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*MultiFactorial[n - k, k]]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 3] + 3^10] &]

for(n=1, 1e3, if(ispseudoprime(prod(i=0, floor((n-1)/3), n-3*i) + 3^10), print1(n, ", "))) \\ Altug Alkan, Nov 18 2015

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

A265200 Numbers n such that n!3 + 3^7 is prime, where n!3 = n!!! is a triple factorial number (A007661).

Original entry on oeis.org

8, 10, 11, 13, 16, 19, 20, 22, 37, 38, 47, 73, 92, 94, 100, 218, 241, 284, 482, 541, 736, 787, 829, 916, 1147, 1312, 1856, 1928, 2035, 3134, 4958, 5503, 8042, 16898, 16987, 24548, 25076, 35086

Offset: 1

Robert Price, Dec 04 2015

Corresponding primes are: 2267, 2467, 3067, 5827, 60427, 1108747, 4190987, 24346507, 664565853954187, ...
a(39) > 50000.
Terms > 38 correspond to probable primes.

			11!3 + 3^7 = 11*8*5*2 + 2187 = 3067 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!3+2187.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A007661, A037082, A084438, A123910, A242994, A261145.

MultiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*MultiFactorial[n - k, k]]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 3] + 3^7] &]
Select[Range[35100],PrimeQ[Times@@Range[#,1,-3]+2187]&] (* Harvey P. Dale, Oct 19 2023 *)

tf(n) = prod(i=0, (n-1)\3, n-3*i);
for(n=1, 1e4, if(ispseudoprime(tf(n) + 3^7), print1(n , ", "))) \\ Altug Alkan, Dec 04 2015

A247463 Numbers n such that n!3 - 3^3 is prime, where n!3 = n!!! is a triple factorial number (A007661).

Original entry on oeis.org

8, 11, 13, 22, 29, 49, 56, 61, 103, 142, 149, 257, 319, 365, 680, 736, 737, 749, 947, 974, 1040, 4277, 4678, 9961, 10652, 15545, 18064, 31325, 34918, 41032

Offset: 1

Robert Price, Sep 17 2014

Large terms correspond to probable primes.

a(31) > 50000.

			11!3-27 = 11*8*5*2-27 = 853 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!3-27
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester
Eric Weisstein's World of Mathematics, Multifactorial

Cf. A007661, A037082, A084438, A243078.

MultiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*MultiFactorial[n - k, k]]];
lst={};Do[If[PrimeQ[MultiFactorial[n, 3] - 27], AppendTo[lst, n]], {n, 100}];lst
Select[Range[6,1100],PrimeQ[Times@@Range[#,1,-3]-27]&] (* Harvey P. Dale, Mar 16 2023 *)

A247464 Numbers n such that n!!! - 3^4 is prime.

Original entry on oeis.org

10, 13, 14, 17, 20, 26, 29, 31, 32, 50, 59, 77, 82, 104, 164, 185, 217, 263, 293, 361, 437, 442, 545, 547, 599, 608, 623, 739, 782, 1081, 1120, 1138, 1429, 2516, 2518, 4277, 4529, 5438, 5596, 11945, 12716, 13955, 14540, 31730, 31769, 42964, 46396

Offset: 1

Robert Price, Sep 17 2014

Large terms correspond to probable primes.

a(48) > 50000.

			10!3-81 = 10*7*4*1-81 = 199 is prime, so 10 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!3-81
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester
Eric Weisstein's World of Mathematics, Multifactorial

Cf. A007661, A037082, A084438, A243078.

MultiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*MultiFactorial[n - k, k]]];
lst={};Do[If[PrimeQ[MultiFactorial[n, 3] - 81], AppendTo[lst, n]], {n, 100}];lst

cp's OEIS Frontend

A084438 Positive integers k such that k!!! - 1 = A007661(k) - 1 is prime.

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A242994 Numbers n such that n!3 - 3 is prime, where n!3 = n!!! is a triple factorial number (A007661).

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A001281 Image of n under the map n->n/2 if n even, n->3n-1 if n odd.

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A037083 Numbers k such that k!!! + 1 is prime (0 is included by convention).

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A243078 Numbers k such that k!3 - 3^2 is prime, where k!3 = k!!! is a triple factorial number (A007661).

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Mathematica

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A261145 Numbers n such that n!3 + 3^10 is prime, where n!3 = n!!! is a triple factorial number (A007661).

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Mathematica

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

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A265200 Numbers n such that n!3 + 3^7 is prime, where n!3 = n!!! is a triple factorial number (A007661).