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A114805 Cumulative sum of quintuple factorial numbers n!!!!! (A085157). 1
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 (list; graph; refs; listen; history; text; internal format)
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: A212365 A131075 A133523 * A196722 A181120 A000601

Adjacent sequences:  A114802 A114803 A114804 * A114806 A114807 A114808

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Feb 18 2006

STATUS

approved

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Last modified September 23 09:57 EDT 2020. Contains 337298 sequences. (Running on oeis4.)