A085157 -id:A085157 - cp's OEIS frontend

A114421 Quintuple primorial n##### = n#5.

1, 2, 3, 5, 7, 11, 26, 51, 95, 161, 319, 806, 1887, 3895, 6923, 14993, 42718, 111333, 237595, 463841, 1064503, 3118414, 8795307, 19720385, 41281849, 103256791, 314959814, 905916621, 2110081195, 4499721541, 11668017383

Offset: 0

Jonathan Vos Post, Feb 12 2006

This is to quintuple factorial A085157 = n!!!!!, as double primorial A079078 = n## is to double factorial A006882 = n!! and as primorial A002110 = n# is to factorial A000142 = n!. There is an obvious generalization to multiprimorial. (n#5)*((n-1)#5)*((n-2)#5)*((n-3)#5)*((n-4)#5) = n#. n#5 is a k-almost prime for k = ceiling(n/5).

			n##### is also written n#5.
0#5 = p(0) = 1.
1#5 = p(1) = 2.
2#5 = p(2) = 3.
3#5 = p(3) = 5.
4#5 = p(4) = 7.
5#5 = p(5)p(0) = 11*1 = 11.
6#5 = p(6)p(1) = 13*2 = 26.
7#5 = p(7)p(2) = 17*3 = 51.
8#5 = p(8)p(3) = 19*5 = 95.
9#5 = p(9)p(4) = 23*7 = 161.
10#5 = p(10)p(5)p(0) = 29*11*1 = 319.
11#5 = p(11)p(6)p(1) = 31*13*2 = 806.
12#5 = 37*17*3 = 1887.
13#5 = 41*19*5 = 3895.
14#5 = 43*23*7 = 6923.
15#5 = 47*29*11*1 = 14993.
16#5 = 53*31*13*2 = 42718.
17#5 = 59*37*17*3 = 111333.
18#5 = 61*41*19*5 = 237595.
19#5 = 67*43*23*7 = 463841.
20#5 = 71*47*29*11*1 = 1064503.
21#5 = 73*53*31*13*2 = 3118414.
22#5 = 79*59*37*17*3 = 8795307.
23#5 = 83*61*41*19*5 = 19720385.
24#5 = 89*67*43*23*7 = 41281849.
25#5 = 97*71*47*29*11*1 = 103256791.
26#5 = 101*73*53*31*13*2 = 314959814.
27#5 = 103*79*59*37*17*3 = 905916621.
28#5 = 107*83*61*41*19*5 = 2110081195.
29#5 = 109*89*67*43*23*7 = 4499721541.
30#5 = 113*97*71*47*29*11*1 = 11668017383.

Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142, A002110, A006882, A007661, A007662, A079078.

a(n) = n##### = prime(n)*((n-5)#####) = Prod[i == n mod 5, to n] prime(i). Notationally, prime(0) = 1; (-n)##### = 0#### = 1.

A114869 s(n) = floor(n^(n/5)/n!!!!!).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 6, 5, 5, 7, 10, 16, 12, 14, 18, 26, 39, 31, 35, 45, 64, 98, 79, 88, 114, 163, 249, 200, 223, 291, 416, 636, 511, 572, 745, 1067, 1634, 1316, 1474, 1922, 2755, 4222, 3405, 3817, 4982, 7147, 10961, 8848, 9925, 12966

Offset: 1

Jonathan Vos Post, Feb 20 2006

This sequence is an approximation of a quintuple factorial analog to Stirling's approximation to factorial. Note that a(n) is exact for n = 1, 5, 10.

			a(10) = floor(10^2/10!!!!!) = floor(10^2/50) = floor(2) = 2.
a(15) = floor(15^3/15!!!!!) = floor((15^3)/750) = floor(4.5) = 4.
a(20) = floor(20^4/20!!!!!) = floor((20^4)/15000) = floor(10.6666667) = 10.
a(25) = floor(25^5/25!!!!!) = floor((25^5)/375000) = floor(26.0416667) = 26.
a(30) = floor(30^6/30!!!!!) = floor((30^6)/11250000) = floor(64.8) = 64.
a(35) = floor(35^7/35!!!!!) = floor((35^7)/393750000) = floor(163.401389) = 163.

Cf. A000312, A006882, A055775, A085157.

fac[n_Integer, m_Integer] := Block[{t = n, f = Max[1, n]}, While[t > m, t -= m; f *= t]; f]; a[n_] := Floor[n^(n/5)/fac[n, 5]]; Array[a, 65] (* Giovanni Resta, Jun 15 2016 *)

a(n) = floor(n^(n/5)/n!!!). a(n) = floor((A000312(n)^(1/5))/A085157(n)).

Corrected and extended by Giovanni Resta, Jun 15 2016

A114968 a(n) = floor(n^(n/6)/n!!!!!!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 6, 4, 4, 5, 7, 10, 15, 11, 12, 14, 18, 26, 37, 28, 29, 35, 46, 64, 94, 71, 74, 89, 117, 163, 238, 180, 188, 226, 297, 416, 608, 461, 481, 580, 763, 1067, 1563, 1187, 1240, 1496, 1969, 2755, 4038, 3070

Offset: 1

Jonathan Vos Post, Feb 22 2006

This sequence is an approximation of a sextuple factorial analog to Stirling's approximation to factorial. Note that a(n) is exact for n = 1, 6, 8, 9, 12.

			a(12) = floor(12^2/12!!!!!) = floor(12^2/72) = floor(2) = 2.
a(18) = floor(18^3/18!!!!!) = floor((18^3)/1296) = floor(4.5) = 4.
a(24) = floor(24^4/20!!!!!) = floor((24^4)/31104) = floor(10.6666667) = 10.
a(30) = floor(30^5/25!!!!!) = floor((30^5)/933120) = floor(26.0416667) = 26.
a(36) = floor(36^6/30!!!!!) = floor((36^6)/33592320) = floor(64.8) = 64.

Cf. A000312, A006882, A055775, A085157, A085158.

fac[n_, m_] := Block[{t = n, f = Max[1, n]}, While[t > m, t -= m; f *= t]; f]; a[n_] := Floor[n^(n/6)/fac[n, 6]]; Array[a, 65] (* Giovanni Resta, Jun 15 2016 *)

a(n) = floor(n^(n/6)/n!!!!!!). a(n) = floor(n^(n/6)/n!6). a(n) = floor((A000312(n)^(1/6))/A085158(n)).

Corrected and extended by Giovanni Resta, Jun 15 2016

cp's OEIS Frontend

A114421 Quintuple primorial n##### = n#5.

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A114869 s(n) = floor(n^(n/5)/n!!!!!).

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A114968 a(n) = floor(n^(n/6)/n!!!!!!).

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