A108895 Partial sums of quadruple factorial numbers n!!!! (A007662).

1, 2, 4, 7, 11, 16, 28, 49, 81, 126, 246, 477, 861, 1446, 3126, 6591, 12735, 22680, 52920, 118755, 241635, 450480, 1115760, 2629965, 5579085, 10800210, 28097490, 68981025, 151556385, 302969010, 821887410, 2089276995, 4731688515

Offset

0,2

Comments

Quadruple factorial numbers n!!!! = n*(n-4)!!!!, 0!!!! = 1!!!! = 1, 2!!!! = 2, 3!!!! = 3. The cumulative sum a(n) is prime for n = 1, 3, 4 and never again, as all values from a(8) = 81 are multiples of 3. The cumulative sum a(n) is semiprime for n = 2, 7 and never again, as all values from a(16) are divisible by both 3 and 5.

References

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

Links

Table of n, a(n) for n=0..32.

Formula

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

a(n) = Sum_{i=0..n} A007662(i).

Example

a(31) = 1 + 1 + 2 + 3 + 4 + 5 + 12 + 21 + 32 + 45 + 120 + 231 + 384 + 585 + 1680 + 3465 + 6144 + 9945 + 30240 + 65835 + 122880 + 208845 + 665280 + 1514205 + 2949120 + 5221125 + 17297280 + 40883535 + 82575360 + 151412625 + 518918400 + 1267389585 = 2089276995 = 3 * 5 * 13 * 337 * 31793.

Mathematica

NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[ Sum[ NFactorialM[i, 4], {i, 0, n}], {n, 0, 33}] (* Robert G. Wilson v, Feb 21 2006 *)

Crossrefs

Cf. A000040, A001358, A007662, A114347.

Sequence in context: A357933 A237821 A120118 * A146929 A146921 A153535

Adjacent sequences: A108892 A108893 A108894 * A108896 A108897 A108898

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 08 2006

Status

approved