A114347 Cumulative sum of triple factorial numbers a(n) = n!!! (A007661).

1, 2, 4, 7, 11, 21, 39, 67, 147, 309, 589, 1469, 3413, 7053, 19373, 48533, 106773, 316213, 841093, 1947653, 6136453, 17158933, 41503253, 137845653, 402385173, 1010993173, 3515895573, 10658462613, 27699486613, 100341656213, 314618667413, 842890411413

Offset: 0

Author

Jonathan Vos Post, Feb 08 2006

Keywords

• easy
• nonn

Comments

Triple factorial numbers n!!! = n*(n-3)!!!, 0!!! = 1!!! = 1, 2!!! = 2.

The cumulative sum a(n) is prime for n = 1, 3, 4, 7, 12, 14, 15, 17, 19.

The cumulative sum a(n) is semiprime for n = 2, 5, 6, 9, 10, 11, 13, 16, 18, 20, 22, 26, 28, 29.

Examples

			a(29) = 1 + 1 + 2 + 3 + 4 + 10 + 18 + 28 + 80 + 162 + 280 + 880 + 1944 + 3640 + 12320 + 29160 + 58240 + 209440 + 524880 + 1106560 + 4188800 + 11022480 + 24344320 + 96342400 + 264539520 + 608608000 + 2504902400 + 7142567040 + 17041024000 + 726421696 = 100341656213 = 79 * 1270147547.
		

References

• J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

Links

• Delbert L. Johnson, Table of n, a(n) for n = 0..1141

Crossrefs

Cf. A000040, A001358, A007661.

Formula

a(n) = Sum_{i=0..n} i!!!.

a(n) = Sum_{i=0..n} A007661(i).