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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)).

%I A088494 #18 Mar 28 2022 07:44:12
%S A088494 15,20,32,36,48,41,64,72,80,78,96,81,112,120,128,120,144,94,160,168,
%T A088494 176,162,192,200,208,216,224,177,240,218,256,264,272,280,288,195,304,
%U A088494 312,320,288,336,261,352,360,368,330,384,392,400,408,416,212,432,440,448
%N 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)).
%C A088494 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:
%C A088494    1,   1,    1,    1,    1,    1,    1,    1, ...
%C A088494    1,   2,    2,    2,    2,    2,    2,    2, ...
%C A088494    1,   3,    6,    6,    6,    6,    6,    6, ...
%C A088494    4,  12,   24,   24,   24,   24,   24,   24, ...
%C A088494    4,  60,  120,  120,  120,  120,  120,  120, ...
%C A088494   24, 360,  720,  720,  720,  720,  720,  720, ...
%C A088494   24, 840, 2520, 5040, 5040, 5040, 5040, 5040, ...
%C A088494 The a(n) are some sort of average integer value of ratios of neighbored rows in the first 8 columns.
%H A088494 G. C. Greubel, <a href="/A088494/b088494.txt">Table of n, a(n) for n = 2..5000</a>
%F A088494 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
%p A088494 P := proc(n,k)
%p A088494     local a,i ;
%p A088494     a := 1 ;
%p A088494     for i from 1 to numtheory[pi](n)/2^(k-1) do
%p A088494         a := ithprime(i) *a ;
%p A088494     end do:
%p A088494     n!/a ;
%p A088494 end proc:
%p A088494 A088494 := proc(n)
%p A088494     add( floor(P(n,k)/P(n-1,k)),k=1..8) ;
%p A088494 end proc: # _R. J. Mathar_, Sep 17 2013
%t A088494 p[n_, k_]:= p[n,k]= n!/Product[Prime[i], {i, PrimePi[n]/2^(k-1)}];
%t A088494 f[n_]:= f[n]= Sum[Floor[p[n, k]/p[n-1, k]], {k,8}];
%t A088494 Table[f[n], {n,2,70}]
%o A088494 (Sage)
%o A088494 @CachedFunction
%o A088494 def f(n,k): return product( nth_prime(j) for j in (1..prime_pi(n)/2^(k-1)) )
%o A088494 def A088494(n): return sum( (n*f(n-1,k)//f(n,k)) for k in (1..8) )
%o A088494 [A088494(n) for n in (2..70)] # _G. C. Greubel_, Mar 27 2022
%Y A088494 Cf. A088140, A088493.
%K A088494 nonn
%O A088494 2,1
%A A088494 _Roger L. Bagula_, Nov 10 2003
%E A088494 Meaningful name by _R. J. Mathar_, Sep 17 2013