A114800 -id:A114800 - cp's OEIS frontend

A204661 Numbers n such that n!8+1 is prime (for n!8 see A114800).

0, 1, 2, 4, 6, 28, 30, 46, 60, 72, 86, 90, 112, 154, 162, 206, 280, 354, 400, 512, 606, 614, 678, 790, 938, 1054, 1092, 1148, 1582, 1788, 2088, 2206, 2598, 2912, 3672, 4642, 6272, 6428, 7084, 7604, 8580, 9464, 12762, 18386, 24910, 30448, 31696, 40288, 41682, 45730

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).

No other terms < 50000. - Robert Price, Jul 29 2012

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!8+1.

Cf. A204657, A204658, A204659, A204660, 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[#, 8] + 1] & ] (* Robert Price, Apr 19 2019 *)
Select[Range[0,46000],PrimeQ[Times@@Range[#,1,-8]+1]&] (* Harvey P. Dale, Apr 12 2022 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)+1)& print1(n","))

a(35)-a(50) from Robert Price, Jul 29 2012

A288715 Primes of the form k!8+1, where k!8 is the octuple factorial number (A114800).

Original entry on oeis.org

2, 3, 5, 7, 26881, 55441, 96909121, 132843110401, 48704929136641, 152349556104345601, 1397121162877440001, 383414179456168545484801, 81419177249980419349301811609600001, 13189906714496827934586893480755200001

Offset: 1

Robert Price, Jun 13 2017

nonn

Robert Price, Table of n, a(n) for n = 1..21
Henri& Renaud Lifchitz, PRP Records.Search for n!8+1.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A204661.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 8] + 1, {i, 0, 100}], PrimeQ[#]&]
Select[Table[Times@@Range[n,1,-8]+1,{n,200}],PrimeQ] (* Harvey P. Dale, Dec 26 2022 *)

A288716 Primes of the form k!8+2, where k!8 is the octuple factorial number (A114800).

Original entry on oeis.org

3, 5, 7, 11, 67, 107, 1367, 2417, 16931, 126227, 592517, 65909027, 3493178327, 7547514977, 14454403427, 385235284982627, 2667042724170827, 98523573068265783062627, 121598818552526868243555286516922298627484453127

Offset: 1

Robert Price, Jun 13 2017

nonn

Robert Price, Table of n, a(n) for n = 1..25
Henri& Renaud Lifchitz, PRP Records.Search for n!8+2.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A204663.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 8] + 2, {i, 0, 100}], PrimeQ[#]&]

A289756 Primes of the form k!8-1, where k!8 is the octuple factorial number (A114800).

Original entry on oeis.org

2, 3, 5, 7, 19, 47, 83, 127, 359, 1847, 26879, 55439, 13366079, 188743679, 38761631999, 9033331507199, 3896394330931199, 152349556104345599, 5305528527460761599, 57299708096576225279999, 160680029832131217407999, 383414179456168545484799

Offset: 1

Robert Price, Jul 11 2017

nonn

Robert Price, Table of n, a(n) for n = 1..30
Henri & Renaud Lifchitz, PRP Records.Search for n!8-1.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A204662.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 8] - 1, {i, 2, 100}], PrimeQ[#]&]

A289759 Primes of the form k!8-2, where k!8 is the octuple factorial number (A114800).

Original entry on oeis.org

2, 3, 5, 7, 31, 103, 151, 3823, 16927, 126223, 137227543, 76663738303, 475493443423, 1132114642884223, 232032717002861773, 494437513909964623, 8949366251999798623, 22043108115271868623, 30822262564469609858623, 29990243754746489604765373

Offset: 1

Robert Price, Jul 11 2017

nonn

Robert Price, Table of n, a(n) for n = 1..32
Henri & Renaud Lifchitz, PRP Records.Search for n!8-2.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A204664.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 8] - 2, {i, 3, 100}], PrimeQ[#]&]

A288095 Decimal expansion of m(8) = Sum_{n>=0} 1/n!8, the 8th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.9892412126901365441336421348019099438359273924576814826209556653...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114800 (n!8), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), this sequence (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(107)); (1/8)*Exp(1/8)*(8 + (&+[8^(k/8)*Gamma(k/8, 1/8): k in [1..7]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[8], 10, 103][[1]]

default(realprecision, 105); (1/8)*exp(1/8)*(8 + sum(k=1,7, 8^(k/8)*(gamma(k/8) - incgam(k/8, 1/8)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/8)*exp(1/8)*(8 + sum(8^(k/8)*(gamma(k/8) - gamma_inc(k/8, 1/8)) for k in (1..7))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A204662 Numbers n such that n!8-1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 14, 16, 18, 22, 28, 30, 42, 48, 58, 68, 80, 86, 92, 108, 110, 112, 130, 198, 220, 230, 322, 432, 460, 478, 686, 706, 714, 842, 950, 1010, 1090, 1314, 1904, 2264, 2804, 3164, 3324, 4740, 4824, 4918, 5086, 5442, 6994, 7898, 8236, 8684, 10088, 13990, 15320, 17570, 18218, 21564, 22198, 22684, 24314, 24780, 25790, 38726

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).

No other terms < 50000. - Robert Price, Aug 15 2012

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!8-1.

Cf. A204657, A204658, A204659, A204660, A204661, 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[#, 8] - 1] & ] (* Robert Price, Apr 19 2019 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)-1)& print1(n","))

a(39)-a(64) from Robert Price, Aug 15 2012

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

Original entry on oeis.org

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

A288327 Decuple factorial, 10-factorial, n!10, n!!!!!!!!!!.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 39, 56, 75, 96, 119, 144, 171, 200, 231, 528, 897, 1344, 1875, 2496, 3213, 4032, 4959, 6000, 7161, 16896, 29601, 45696, 65625, 89856, 118881, 153216, 193401, 240000, 293601, 709632, 1272843, 2010624, 2953125, 4133376

Offset: 0

Robert Price, Jun 07 2017

			a(13) = 13 * 3 * 1 = 39.

Robert Price, Table of n, a(n) for n = 0..803
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A006852, A007661, A007662, A085157, A085158, A114799, A114800, A114806, A342033.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-10);
    fi;
  end;
List([0..50], n-> a(n) ); # G. C. Greubel, Aug 22 2019

b:=func< n | n le 10 select n else n*Self(n-10) >;
[1] cat [b(n): n in [1..50]]; // G. C. Greubel, Aug 22 2019

a:= n-> `if`(n<1, 1, n*a(n-10)); seq(a(n), n=0..50); # G. C. Greubel, Aug 22 2019

MultiFactorial[n_, k_]:=If[n<1, 1 ,n*MultiFactorial[n-k, k]];
Table[MultiFactorial[i, 10], {i, 0, 100}]
Table[Times@@Range[n,1,-10],{n,0,50}] (* Harvey P. Dale, Aug 11 2019 *)

a(n)=if(n<1, 1, n*a(n-10));
vector(40, n, n--; a(n) ) \\ G. C. Greubel, Aug 22 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-10)
[a(n) for n in (0..50)] # G. C. Greubel, Aug 22 2019

a(n)=1 for n < 1, otherwise a(n) = n*a(n-10).

Sum_{n>=0} 1/a(n) = A342033. - Amiram Eldar, May 23 2022

cp's OEIS Frontend

A204661 Numbers n such that n!8+1 is prime (for n!8 see A114800).

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A288715 Primes of the form k!8+1, where k!8 is the octuple factorial number (A114800).

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A288716 Primes of the form k!8+2, where k!8 is the octuple factorial number (A114800).

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A289756 Primes of the form k!8-1, where k!8 is the octuple factorial number (A114800).

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A289759 Primes of the form k!8-2, where k!8 is the octuple factorial number (A114800).

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A288095 Decimal expansion of m(8) = Sum_{n>=0} 1/n!8, the 8th reciprocal multifactorial constant.

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A204662 Numbers n such that n!8-1 is prime.

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

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