A088493 -id:A088493 - cp's OEIS frontend

A088491 a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).

2, 3, 4, 5, 3, 7, 4, 9, 3, 11, 3, 13, 3, 15, 4, 17, 3, 19, 3, 21, 3, 23, 3, 25, 3, 27, 3, 29, 3, 31, 4, 33, 3, 35, 3, 37, 3, 39, 3, 41, 3, 43, 3, 45, 3, 47, 3, 49, 3, 51, 3, 53, 3, 55, 3, 57, 3, 59, 3, 61, 3, 63, 4, 65, 3, 67, 3, 69, 3, 71, 3, 73, 3, 75, 3, 77, 3, 79, 3, 81, 3, 83, 3, 85, 3, 87

Offset: 2

Roger L. Bagula, Nov 10 2003

nonn

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

Cf. A004001, A088493.

Conway[n_]:= Conway[n]= If[n<3, 1, Conway[Conway[n-1]] +Conway[n-Conway[n-1]]];
f[n_]:= f[n]= Product[Conway[i], {i,Floor[n/2]}];
a[n_]:= a[n]= Floor[n*f[n-1]/f[n]];
Table[a[n], {n, 2, 100}] (* modified by G. C. Greubel, Mar 27 2022 *)

@CachedFunction
def b(n): # A004001
    if (n<3): return 1
    else: return b(b(n-1)) + b(n-b(n-1))
def f(n): return product( b(j) for j in (1..(n//2)) )
def A088491(n): return (n*f(n-1)//f(n))
[A088491(n) for n in (2..100)] # G. C. Greubel, Mar 27 2022

a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).

Edited by G. C. Greubel, Mar 27 2022

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)).

Original entry on oeis.org

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

Roger L. Bagula, Nov 10 2003

nonn

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.

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

Cf. A088140, A088493.

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

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}]

@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

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

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

cp's OEIS Frontend

A088491 a(n) = floor(p(n)/p(n-1)), where p(n) = n!/(Product_{j=1..floor(n/2)} A004001(j)).

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