A007662 -id:A007662 - cp's OEIS frontend

A114420 Quadruple primorial n#### = n#4.

1, 2, 3, 5, 7, 22, 39, 85, 133, 506, 1131, 2635, 4921, 20746, 48633, 123845, 260813, 1224014, 2966613, 8297615, 18517723, 89353022, 234362427, 688702045, 1648077347, 8667243134, 23670605127, 70936310635, 176344276129

Offset: 0

Jonathan Vos Post, Feb 12 2006

This is to quadruple factorial A007662 = 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####)*((n-1)####)*((n-2)####)*((n-3)####) = n#. n#### is a k-almost prime for k = ceiling(n/4).

			n#### is also written n#4.
0#### = p(0) = 1.
1#### = p(1) = 2.
2#### = p(2) = 3.
3#### = p(3) = 5.
4#### = p(4)p(0) = 7*1 = 7.
5#### = p(5)p(1) = 11*2 = 22.
6#### = p(6)p(2) = 13*3 = 39.
7#### = p(7)p(3) = 17*5 = 85.
8#### = p(8)p(4)p(0) = 19*7*1 = 133.
9#### = p(9)p(5)p(1) = 23*11*2 = 506.
10#### = p(10)p(6)p(2) = 29*13*3 = 1131.
11#### = p(11)p(7)p(3) = 31*17*5 = 2635.
12#### = 37*19*7*1 = 4921.
13#### = 41*23*11*2 = 20746.
14#### = 43*29*13*3 = 48633.
15#### = 47*31*17*5 = 123845.
16#### = 53*37*19*7*1 = 260813.
17#### = 59*41*23*11*2 = 1224014.
18#### = 61*43*29*13*3 = 2966613.
19#### = 67*47*31*17*5 = 8297615.
20#### = 71*53*37*19*7*1 = 18517723.
21#### = 73*59*41*23*11*2 = 89353022.
22#### = 79*61*43*29*13*3 = 234362427.
23#### = 83*67*47*31*17*5 = 688702045.
24#### = 89*71*53*37*19*7*1 = 1648077347.
25#### = 97*73*59*41*23*11*2 = 8667243134.
26#### = 101*79*61*43*29*13*3 = 23670605127.
27#### = 103*83*67*47*31*17*5 = 70936310635.
28#### = 107*89*71*53*37*19*7*1 = 176344276129.

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-4)####) = Prod[i == n mod 4, to n] prime(i). Notationally, prime(0) = 1; (-n)#### = 0#### = 1.

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

Original entry on oeis.org

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.

A114868 a(n) = floor(n^(n/4)/n!!!!).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 3, 2, 3, 4, 7, 6, 7, 10, 17, 14, 18, 26, 41, 36, 44, 64, 104, 91, 112, 163

Offset: 1

Jonathan Vos Post, Feb 20 2006

This sequence is an approximation to a quadruple factorial analog of Stirling's approximation to the factorial function. Note that a(n) is exact for n = 1, 4, 8.

			a(8) = floor((8^2)/8!!!!) = floor((8^2)/32) = floor(2) = 2.
a(9) = floor((9^2.25)/9!!!!) = floor((9^2.25)/45) = floor(3.11769145) = 3.
a(16) = floor((16^4)/16!!!!) = floor((16^4)/6144) = floor(10.6666667) = 10.
a(20) = floor((20^5)/20!!!!) = floor((20^5)/122880) = floor(26.0416667) = 26.

Cf. A000312, A006882, A055775, A007662.

a(n) = floor(n^(n/4)/n!!!). a(n) = floor((A000312(n)^(1/4))/A007662(n)).

cp's OEIS Frontend

A114420 Quadruple primorial n#### = n#4.

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

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A114868 a(n) = floor(n^(n/4)/n!!!!).

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