A079078 -id:A079078 - cp's OEIS frontend

A114410 Cumulative sum of double primorials (A079078).

1, 3, 6, 16, 37, 147, 420, 2290, 7477, 50487, 200910, 1534220, 7099871, 61765581, 301088574, 2870376944, 15554495573, 167142509403, 940873745772, 11097270672382, 66032188454581, 807449164097111, 5147307668890832

Offset: 0

Jonathan Vos Post, Feb 12 2006

The cumulative sum is prime for a(2) = 3, a(4) = 37, a(8) = 7477, a(12) = 7099871, a(16) = 15554495573. The sum a(n) is semiprime for n = 2, 9.

			n 0## + ... + n##
0 1.
1 1+2 = 3.
2 1+2+3 = 6.
3 1+2+3+10 = 16.
4 1+2+3+10+21 = 37.
5 1+2+3+10+21+110 = 147.
6 1+2+3+10+21+110+273 = 420.
7 1+2+3+10+21+110+273+1870 = 2290.
8 1+2+3+10+21+110+273+1870+5187 = 7477.
9 1+2+3+10+21+110+273+1870+5187+ 43010 = 50487.
10 1+2+3+10+21+110+273+1870+5187+ 43010 + 150423 = 200910.

Cf. A079078.

p[0]=1; p[1]=2; p[n_] := p[n] = Prime[n]*p[n - 2]; Accumulate[p /@ Range[0, 22]] (* Giovanni Resta, Jun 14 2016 *)

a(n) = 0## + 1## + ... + n##, where n## = p(n)*(n-2)##, where p(n) is the n-th prime.

Data corrected by Giovanni Resta, Jun 14 2016

A114411 Triple primorial n### = n#3.

Original entry on oeis.org

1, 2, 3, 5, 14, 33, 65, 238, 627, 1495, 6902, 19437, 55315, 282982, 835791, 2599805, 14998046, 49311669, 158588105, 1004869082, 3501128499, 11576931665, 79384657478, 290593665417, 1030346918185, 7700311775366, 29349960207117

Offset: 0

Jonathan Vos Post, Feb 12 2006

nonn

This is to triple factorial A007661 = 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#. n### is a k-almost prime for k = ceiling(n/3).

			n### is also written n#3.
0### = p(0) = 1.
1### = p(1) = 2.
2### = p(2) = 3.
3### = p(3)p(0) = 5*1 = 5.
4### = p(4)p(1) = 7*2 = 14.
5### = p(5)p(2) = 11*3 = 33.
6### = p(6)p(3)p(0) = 13*5*1 = 65.
7### = p(7)p(4)p(1) = 17*7*2 = 238.
8### = p(8)p(5)p(2) = 19*11*3 = 627.
9### = p(9)p(6)p(3)p(0) = 23*13*5*1 = 1495.
10### = p(10)p(7)p(4)p(1) = 29*17*7*2 = 6902.
11### = p(11)p(8)p(5)p(2) = 31*19*11*3 = 19437.
12### = 37*23*13*5*1 = 55315.
13### = 41*29*17*7*2 = 282982.
14### = 43*31*19*11*3 = 835791.
15### = 47*37*23*13*5*1 = 2599805.
27### = 106125732573055 = 5 * 13 * 23 * 37 * 47 * 61 * 73 * 89 * 103.

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

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

a[0] = 1; a[1] = 2; a[2] = 3;
a[n_] := a[n - 3] * Prime[n]
Array[a, 27, 0] (* Jon Maiga, Aug 04 2019 *)

a(n) = n### = prime(n)*((n-3)###) = Prod[i == n mod 3, to n] prime(i). Notationally, prime(0) = 1; (-n)### = 0### = 1.

A213005 a(0)=1, a(n) = least k > a(n-1) such that k*a(n-1) is a triangular number.

Original entry on oeis.org

1, 3, 5, 9, 17, 33, 45, 72, 143, 152, 303, 420, 451, 603, 952, 1398, 1572, 2408, 3762, 4233, 5880, 6325, 8469, 13384, 20079, 34189, 62769, 82665, 87448, 161037, 287283, 371337, 515745, 533505, 573815, 734484, 737035, 737149, 767505, 825495, 887865, 1136468, 2272935

Offset: 0

Alex Ratushnyak, Aug 03 2012

Corresponding triangular numbers t(n)=a(n)*a(n+1): 3, 15, 45, 153, 561, 1485, 3240, 10296, 21736, 46056, 127260, 189420, 271953, 574056, 1330896, 2197656, 3785376, 9058896, 15924546, 24890040, 37191000, ...

Cf. A000217, A214961.
Cf. A081976 (a(0)=1, a(n) = least k > a(n-1) such that k*a(n-1) is a Fibonacci number).
Cf. A006882 (a(0)=a(1)=1, a(n) = least k > a(n-1) such that k*a(n-1) is a factorial).
Cf. A079078 (a(0)=1, a(n) = least k > a(n-1) such that k*a(n-1) is a primorial).

a[0] = 1; a[n_] := a[n] = For[k = a[n-1]+1, True, k++, If[ IntegerQ[ Sqrt[8k*a[n-1]+1] ], Return[k] ] ]; Table[ Print[a[n]]; a[n], {n, 0, 42}] (* Jean-François Alcover, Sep 14 2012 *)

a = 1
for n in range(55):
    print(a, end=',')
    b = k = 0
    while k<=a:
        tn = b*(b+1)//2
        k = 0
        if tn%a==0:
            k = tn // a
        b += 1
    a = k

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

Original entry on oeis.org

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.

A114995 Numbers k such that the sum of the first k double factorials (1!! + 2!! + ... + k!!) is prime.

Original entry on oeis.org

2, 5, 9, 10, 13, 14, 22, 25, 50, 66, 125, 250, 293, 314, 349, 1622, 1642, 2937, 2966, 3841, 4298, 4898, 5270

Offset: 1

Giovanni Resta, Feb 23 2006

nonn

a(18) must be greater than 2300. The sum of the first 1642 double factorials is a (probable) prime with 2286 digits.

Numbers corresponding to a(18)-a(23) are probable primes. - Farideh Firoozbakht, May 16 2010

			5 belongs since 1!! + 2!! + 3!! + 4!! + 5!! = 29, a prime.

Cf. A114410, A079078.

a(18)-a(23) from Farideh Firoozbakht, May 16 2010

cp's OEIS Frontend

A114410 Cumulative sum of double primorials (A079078).

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A114411 Triple primorial n### = n#3.

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A213005 a(0)=1, a(n) = least k > a(n-1) such that k*a(n-1) is a triangular number.

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

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

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A114995 Numbers k such that the sum of the first k double factorials (1!! + 2!! + ... + k!!) is prime.

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