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A015650 Number of ordered 5-tuples of integers from [ 1..n ] with no global factor.

%I A015650 #27 Jun 08 2025 05:02:11
%S A015650 1,5,19,49,118,225,434,729,1209,1850,2850,4059,5878,8044,11020,14566,
%T A015650 19410,24789,32103,40213,50615,62260,77209,93099,113504,135431,162341,
%U A015650 191396,227355,264463,310838,359322,417212,478408,551944,626971,718360,812311,922407,1036667
%N A015650 Number of ordered 5-tuples of integers from [ 1..n ] with no global factor.
%H A015650 Chai Wah Wu, <a href="/A015650/b015650.txt">Table of n, a(n) for n = 1..10000</a>
%F A015650 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^5. - _Ilya Gutkovskiy_, Feb 14 2020
%F A015650 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
%F A015650 a(n) ~ n^5 / (120*zeta(5)). - _Amiram Eldar_, Jun 08 2025
%t A015650 a[n_] := Sum[DivisorSum[k, MoebiusMu[k/#]*Binomial[# + 3, 4] &], {k, 1, n}]; Array[a, 40] (* _Amiram Eldar_, Jun 07 2025 *)
%o A015650 (Python)
%o A015650 from functools import lru_cache
%o A015650 @lru_cache(maxsize=None)
%o A015650 def A015650(n):
%o A015650     if n == 0:
%o A015650         return 0
%o A015650     c, j = n+1, 2
%o A015650     k1 = n//j
%o A015650     while k1 > 1:
%o A015650         j2 = n//k1 + 1
%o A015650         c += (j2-j)*A015650(k1)
%o A015650         j, k1 = j2, n//j2
%o A015650     return n*(n+1)*(n+2)*(n+3)*(n+4)//120-c+j # _Chai Wah Wu_, Apr 18 2021
%o A015650 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+3, 4))); \\ _Seiichi Manyama_, Jun 12 2021
%o A015650 (PARI) a(n) = binomial(n+4, 5)-sum(k=2, n, a(n\k)); \\ _Seiichi Manyama_, Jun 12 2021
%o A015650 (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 A015650 Column k=5 of A177976.
%Y A015650 Partial sums of A117109.
%Y A015650 Cf. A000389, A002088, A013663, A015631, A015634.
%K A015650 nonn
%O A015650 1,2
%A A015650 _Olivier Gérard_