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

Author

Jonathan Vos Post, Feb 18 2006

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A007662, A085157, A114347.

Programs

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

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

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 *)

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

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

Formula

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

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

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