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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)).
1
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
2,1
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
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
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}]
PROG
(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
CROSSREFS
Sequence in context: A074236 A086770 A111200 * A109659 A294149 A065148
KEYWORD
nonn
AUTHOR
Roger L. Bagula, Nov 10 2003
EXTENSIONS
Meaningful name by R. J. Mathar, Sep 17 2013
STATUS
approved