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Number of ordered 5-tuples of integers from [ 1..n ] with no global factor.
6

%I #21 Jun 12 2021 09:06:10

%S 1,5,19,49,118,225,434,729,1209,1850,2850,4059,5878,8044,11020,14566,

%T 19410,24789,32103,40213,50615,62260,77209,93099,113504,135431,162341,

%U 191396,227355,264463,310838,359322,417212,478408,551944,626971

%N Number of ordered 5-tuples of integers from [ 1..n ] with no global factor.

%H Chai Wah Wu, <a href="/A015650/b015650.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^5. - _Ilya Gutkovskiy_, Feb 14 2020

%F a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)/120 - Sum_{j=2..n} a(floor(n/j)) = A000389(n+4) - Sum_{j=2..n} a(floor(n/j)). - _Chai Wah Wu_, Apr 18 2021

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A015650(n):

%o if n == 0:

%o return 0

%o c, j = n+1, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c += (j2-j)*A015650(k1)

%o j, k1 = j2, n//j2

%o return n*(n+1)*(n+2)*(n+3)*(n+4)//120-c+j # _Chai Wah Wu_, Apr 18 2021

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+3, 4))); \\ _Seiichi Manyama_, Jun 12 2021

%o (PARI) a(n) = binomial(n+4, 5)-sum(k=2, n, a(n\k)); \\ _Seiichi Manyama_, Jun 12 2021

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^5)/(1-x)) \\ _Seiichi Manyama_, Jun 12 2021

%Y Column k=5 of A177976.

%Y Cf. A000389, A002088, A015631, A015634, A117109.

%K nonn

%O 1,2

%A _Olivier GĂ©rard_