A114805 Cumulative sum of quintuple factorial numbers n!!!!! (A085157).

1, 2, 4, 7, 11, 16, 22, 36, 60, 96, 146, 212, 380, 692, 1196, 1946, 3002, 5858, 11474, 21050, 36050, 58226, 121058, 250226, 480050, 855050, 1431626, 3128090, 6744794, 13409690, 24659690, 42533546, 96820394, 216171626, 442778090, 836528090

Offset

0,2

Comments

a(1) = 2 is prime; a(3) = 7 is prime; a(4) = 11 is prime; and there are no more primes in the sequence. Semiprime values are: a(2) = 4 = 2^2, a(6) = 22, a(10) = 146 = 2 * 73, a(18) = 11474 = 2 * 5737, a(23) = 250226 = 2 * 125113.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{j=0..n} j!5.

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

a(n) = Sum_{j=0..n} A085157(j).

Example

a(10) = 0!5 + 1!5 + 2!5 + 3!5 + 4!5 + 5!5 + 6!5 + 7!5 + 8!5 + 9!5 + 10!5 =
1 + 1 + 2 + 3 + 4 + 5 + 6 + 14 + 24 + 36 + 50 = 146 = 2 * 73.

Maple

b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> sum(b(j), j = 0..n); seq(a(n), n = 0..40); # G. C. Greubel, Aug 21 2019

Mathematica

f5[0]=1; f5[n_]:= f5[n]= If[n<=6, n, n f5[n-5]]; Accumulate[f5/@Range[0, 35]] (* Giovanni Resta, Jun 15 2016 *)

Prog

(PARI) b(n)=if(n<1, 1, n*b(n-5));
vector(40, n, n--; sum(j=0, n, b(j)) ) \\ G. C. Greubel, Aug 21 2019
(Magma) b:= func< n | n eq 0 select 1 else (n lt 6) select n else n*Self(n-5) >;
[(&+[b(j): j in [0..n]]): n in [0..40]]; // G. C. Greubel, Aug 21 2019
(Sage)
@CachedFunction
def b(n):
    if (n<1): return 1
    else: return n*b(n-5)
[sum(b(j) for j in (0..n)) for n in (0..40)] # G. C. Greubel, Aug 21 2019
(GAP)
b:= function(n)
    if n<1 then return 1;
    else return n*b(n-5);
    fi;
  end;
List([0..40], n-> Sum([0..n], j-> b(j)) ); # G. C. Greubel, Aug 21 2019

Crossrefs

Cf. A007662, A085157, A114347.

Sequence in context: A131075 A365698 A133523 * A196722 A181120 A000601

Adjacent sequences: A114802 A114803 A114804 * A114806 A114807 A114808

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 18 2006

Status

approved