A088494 Let P(n,k) = n!/(Product_{i=1..pi(n)/2^(k-1)} prime(i)) be an integer matrix of "partial" factorials. Then a(n) = sum_{k=1..8} floor( P(n,k)/P(n-1,k)).

15, 20, 32, 36, 48, 41, 64, 72, 80, 78, 96, 81, 112, 120, 128, 120, 144, 94, 160, 168, 176, 162, 192, 200, 208, 216, 224, 177, 240, 218, 256, 264, 272, 280, 288, 195, 304, 312, 320, 288, 336, 261, 352, 360, 368, 330, 384, 392, 400, 408, 416, 212, 432, 440, 448

Offset: 2

Author

Roger L. Bagula, Nov 10 2003

Keywords

• nonn

Comments

The auxiliary integer array P is n! divided by the product of the first primes with an upper limit of the prime index given by A000720(n)/2^(k-1). It starts in row n=1 with columns k>=1 as:

1, 1, 1, 1, 1, 1, 1, 1, ...

1, 2, 2, 2, 2, 2, 2, 2, ...

1, 3, 6, 6, 6, 6, 6, 6, ...

4, 12, 24, 24, 24, 24, 24, 24, ...

4, 60, 120, 120, 120, 120, 120, 120, ...

24, 360, 720, 720, 720, 720, 720, 720, ...

24, 840, 2520, 5040, 5040, 5040, 5040, 5040, ...

The a(n) are some sort of average integer value of ratios of neighbored rows in the first 8 columns.

Links

• G. C. Greubel, Table of n, a(n) for n = 2..5000

Crossrefs

Cf. A088140, A088493.

Programs

Maple

P := proc(n,k)
    local a,i ;
    a := 1 ;
    for i from 1 to numtheory[pi](n)/2^(k-1) do
        a := ithprime(i) *a ;
    end do:
    n!/a ;
end proc:
A088494 := proc(n)
    add( floor(P(n,k)/P(n-1,k)),k=1..8) ;
end proc: # R. J. Mathar, Sep 17 2013

Mathematica

p[n_, k_]:= p[n,k]= n!/Product[Prime[i], {i, PrimePi[n]/2^(k-1)}];
f[n_]:= f[n]= Sum[Floor[p[n, k]/p[n-1, k]], {k,8}];
Table[f[n], {n,2,70}]

Sage

@CachedFunction
def f(n,k): return product( nth_prime(j) for j in (1..prime_pi(n)/2^(k-1)) )
def A088494(n): return sum( (n*f(n-1,k)//f(n,k)) for k in (1..8) )
[A088494(n) for n in (2..70)] # G. C. Greubel, Mar 27 2022

Formula

a(n) = Sum_{k=1..8} floor(p(n,k)/p(n-1,k)), where p(n, k) = n!/( Product_{j=1..PrimePi(n)/2^(k-1)} Prime(j) ). - G. C. Greubel, Mar 27 2022

Extensions

Meaningful name by R. J. Mathar, Sep 17 2013