A085158 -id:A085158 - cp's OEIS frontend

A114796 Cumulative product of sextuple factorial A085158.

1, 1, 2, 6, 24, 120, 720, 5040, 80640, 2177280, 87091200, 4790016000, 344881152000, 31384184832000, 7030057402368000, 2847173247959040000, 1822190878693785600000, 1703748471578689536000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!6 * 2!6 * 3!6 * 4!6 * 5!6 * 6!6 * 7!6 * 8!6 * 9!6 * 10!6
= 1 * 2 * 3 * 4 * 5 * 6 * 7 * 16 * 27 * 40 = 87091200 = 2^11 * 3^5 * 5^2 * 7.
Note that a(10) + 1 = 87091201 is prime, as is a(9) + 1 = 2177281.

G. C. Greubel, Table of n, a(n) for n = 0..91
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000178, A006882, A007662, A085150, A085157, A085158, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-6);
    fi;
  end;
List([0..20], n-> Product([0..n], j-> b(j)) ); # G. C. Greubel, Aug 22 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [(&*[b(j): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 22 2019

b:= n-> `if`(n<1, 1, n*b(n-5)); a:= n-> product(b(j), j = 0..n); seq(a(n), n = 0..20); # G. C. Greubel, Aug 22 2019

b[n_]:= b[n]= If[n<1, 1, n*b[n-6]]; a[n_]:= Product[b[j], {j,0,n}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Aug 22 2019 *)

b(n)=if(n<1, 1, n*b(n-6));
vector(20, n, n--; prod(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 22 2019

def b(n):
    if (n<1): return 1
    else: return n*b(n-6)
[product(b(j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 22 2019

a(n) = Product_{j=0..n} j!!!!!!.
a(n) = Product_{j=0..n} j!6.
a(n) = Product_{j=0..n} A085158(j).
a(n) = n!!!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
a(n) = n*(n-6)!!!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.

A279646 Numbers k such that k!6 - 3 is prime, where k!6 is the sextuple factorial number (A085158).

Original entry on oeis.org

5, 6, 8, 10, 68, 82, 92, 98, 118, 286, 796, 878, 1360, 1502, 1516, 1568, 1646, 3628, 3716, 4048, 7982, 12776, 18070, 20594, 29902, 39632, 52988, 53864, 55610, 67448, 85402, 89762

Offset: 1

Robert Price, Jul 07 2017

Corresponding primes are: 2, 3, 13, 37, 73569236156415997, ...
a(33) > 10^5.
Terms > 10 correspond to probable primes.

			10!6 - 3 = 10*4 - 3 = 37 is prime, so 10 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6-3.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[4, 50000], PrimeQ[MultiFactorial[#, 6] - 3] &]

a(27)-a(32) from Robert Price, Aug 03 2018

A283485 Numbers k such that k![6]-2 is prime, where k![6] = A085158 (k) = sextuple factorial.

Original entry on oeis.org

4, 5, 7, 11, 13, 23, 25, 31, 33, 37, 59, 63, 91, 157, 265, 267, 327, 539, 555, 621, 715, 921, 979, 1633, 1821, 2259, 2697, 2809, 2863, 2935, 4213, 4351, 5937, 6885, 8743, 10761, 15159, 17685, 52075, 55147, 68677, 99655

Offset: 1

Robert Price, Mar 08 2017

nonn

a(43) > 10^5.

The first 10 primes associated with this sequence: 2, 3, 5, 53, 89, 21503, 43223, 1339973, 7577953, 49579073.

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. A076185, A085158, A094144, A204657.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[2, 50000], PrimeQ[MultiFactorial[#, 6] - 2] &]
Select[Range[100000],PrimeQ[Times@@Range[#,1,-6]-2]&] (* Harvey P. Dale, Feb 23 2023 *)

a(39)-a(42) from Robert Price, Jul 09 2017

A287207 Numbers k such that k![6] + 2 is prime, where k![6] = A085158(k) = sextuple factorial.

Original entry on oeis.org

0, 1, 3, 5, 9, 17, 27, 45, 51, 53, 93, 197, 213, 221, 245, 279, 845, 927, 2055, 2895, 3615, 5613, 12753, 15737, 17813, 18545, 22629, 47859, 48797

Offset: 1

Robert Price, May 21 2017

nonn

a(30) > 50000.

The first 7 primes associated with this sequence: 3, 3, 5, 7, 29, 937, 229637.

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. A076185, A085158, A094144, A204657.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 2] &]
Select[Range[0,10000],PrimeQ[Times@@Range[#,1,-6]+2]&] (* The program generates the first 22 terms of the sequence. *) (* Harvey P. Dale, Dec 27 2022 *)

A287844 Numbers k such that k!6 + 3 is prime, where k!6 is the sextuple factorial number (A085158 ).

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 34, 70, 164, 346, 398, 902, 938, 1426, 1682, 1928, 3596, 3796, 15058, 25654, 37330

Offset: 1

Robert Price, Jun 01 2017

Corresponding primes are: 5, 7, 19, 43, 227, 643, 4483, 14083, 116483, 13404163, ...
a(25) > 50000.
Terms > 34 correspond to probable primes.

			10!6 + 3 = 10*4 + 3 = 43 is prime, so 10 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6+3.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 3] &]
Select[Range[3800],PrimeQ[Times@@Range[#,1,-6]+3]&] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, May 23 2025 *)

A287914 Numbers k such that k!6 + 4 is prime, where k!6 is the sextuple factorial number (A085158 ).

Original entry on oeis.org

1, 3, 7, 9, 11, 15, 19, 27, 35, 59, 71, 75, 95, 109, 153, 155, 169, 189, 277, 355, 383, 405, 455, 625, 843, 853, 879, 1389, 1423, 1515, 1871, 2059, 2677, 3095, 4473, 5691, 5927, 8149, 10789, 12171, 14683, 26383, 34227, 40945

Offset: 1

Robert Price, Jun 02 2017

Corresponding primes are: 5, 7, 11, 31, 59, 409, 1733, 229639, 21827579, ...
a(45) > 50000.
Terms > 35 correspond to probable primes.

			11!6 + 4 = 11*5 + 4 = 59 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6+4.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 4] &]

A287956 Numbers k such that k!6 + 6 is prime, where k!6 is the sextuple factorial number (A085158 ).

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 37, 43, 55, 107, 139, 149, 157, 211, 223, 343, 373, 409, 523, 12049, 16457, 17143, 17543, 18391, 25829, 25945, 31307, 34601, 41687

Offset: 1

Robert Price, Jun 03 2017

Corresponding primes are: 7, 11, 13, 61, 97, 941, 49579081, 2131900231, 5745471106381, ...
a(43) > 50000.
Terms > 35 correspond to probable primes.

			11!6 + 6 = 11*5 + 6 = 61 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6+6.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 6] &]

A288152 Numbers k such that k!6 + 8 is prime, where k!6 is the sextuple factorial number (A085158 ).

Original entry on oeis.org

3, 5, 21, 29, 41, 65, 243, 305, 389, 509, 819, 1653, 7493, 8613, 8619, 10257, 11829, 12977, 15651, 24341, 29367, 31629, 40173

Offset: 1

Robert Price, Jun 05 2017

Corresponding primes are: 11, 13, 8513, 623653, 894930583, 8549258359016383, ...
a(24) > 50000.
Terms > 41 correspond to probable primes.

			21!6 + 8 = 21*15*9*3 + 8 = 8513 is prime, so 21 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6+8.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 8] &]

A288154 Numbers k such that k!6 + 9 is prime, where k!6 is the sextuple factorial number (A085158 ).

Original entry on oeis.org

2, 4, 14, 28, 34, 46, 50, 52, 86, 100, 106, 140, 166, 170, 208, 242, 338, 344, 412, 1360, 2024, 2948, 3650, 5608, 5744, 7618, 8410, 8834, 11872, 12514, 13636, 18742, 20846, 29750, 31312

Offset: 1

Robert Price, Jun 05 2017

Corresponding primes are: 11, 13, 233, 394249, 13404169, 24663654409, 311607296009, ...
a(36) > 50000.
Terms > 50 correspond to probable primes.

			14!6 + 9 = 14*8*2 + 9 = 233 is prime, so 14 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6+9.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 9] &]

A288155 Numbers k such that k!6 + 12 is prime, where k!6 is the sextuple factorial number (A085158 ).

Original entry on oeis.org

0, 1, 5, 7, 11, 13, 17, 19, 23, 25, 41, 67, 71, 101, 109, 151, 163, 181, 233, 241, 265, 355, 433, 563, 767, 997, 1465, 1681, 1861, 1913, 2411, 2539, 2777, 13433, 22355, 30895, 44605

Offset: 1

Robert Price, Jun 05 2017

Corresponding primes are: 13, 13, 17, 19, 67, 103, 947, 1741, 21517, 43237, 894930587, ...
a(37) > 50000.
Terms > 41 correspond to probable primes.

			11!6 + 12 = 11*5 + 12 = 67 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!6+12.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

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

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 6] + 12] &]
Select[Range[0, 45000], PrimeQ[Times @@ Range[#, 1, -6] + 12] &] (* Harvey P. Dale, Jul 09 2020 *)

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A114796 Cumulative product of sextuple factorial A085158.

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A279646 Numbers k such that k!6 - 3 is prime, where k!6 is the sextuple factorial number (A085158).

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A283485 Numbers k such that k![6]-2 is prime, where k![6] = A085158 (k) = sextuple factorial.

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A287207 Numbers k such that k![6] + 2 is prime, where k![6] = A085158(k) = sextuple factorial.

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A287844 Numbers k such that k!6 + 3 is prime, where k!6 is the sextuple factorial number (A085158 ).

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A287914 Numbers k such that k!6 + 4 is prime, where k!6 is the sextuple factorial number (A085158 ).

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A287956 Numbers k such that k!6 + 6 is prime, where k!6 is the sextuple factorial number (A085158 ).

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A288152 Numbers k such that k!6 + 8 is prime, where k!6 is the sextuple factorial number (A085158 ).

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