cp's OEIS Frontend

A015634 Number of ordered quadruples of integers from [ 1..n ] with no global factor.

%I A015634 #31 Jun 08 2025 04:49:24
%S A015634 1,4,13,29,63,106,189,289,444,626,911,1203,1657,2130,2766,3462,4430,
%T A015634 5359,6688,7992,9670,11405,13704,15840,18730,21548,25037,28521,33015,
%U A015634 37067,42522,47690,53940,60108,67760,74748,83886,92433,102629,112469,124809,135763,149952
%N A015634 Number of ordered quadruples of integers from [ 1..n ] with no global factor.
%H A015634 Chai Wah Wu, <a href="/A015634/b015634.txt">Table of n, a(n) for n = 1..10000</a>
%F A015634 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - x^k)^4. - _Ilya Gutkovskiy_, Feb 14 2020
%F A015634 a(n) = n*(n+1)*(n+2)*(n+3)/24 - Sum_{j=2..n} a(floor(n/j)) = A000332(n+3) - Sum_{j=2..n} a(floor(n/j)). - _Chai Wah Wu_, Apr 18 2021
%F A015634 a(n) ~ n^4 / (24*zeta(4)). - _Amiram Eldar_, Jun 08 2025
%t A015634 a[n_] := Sum[DivisorSum[k, MoebiusMu[k/#]*Binomial[# + 2, 3] &], {k, 1, n}]; Array[a, 45] (* _Amiram Eldar_, Jun 07 2025 *)
%o A015634 (Python)
%o A015634 from functools import lru_cache
%o A015634 @lru_cache(maxsize=None)
%o A015634 def A015634(n):
%o A015634     if n == 0:
%o A015634         return 0
%o A015634     c, j = 1, 2
%o A015634     k1 = n//j
%o A015634     while k1 > 1:
%o A015634         j2 = n//k1 + 1
%o A015634         c += (j2-j)*A015634(k1)
%o A015634         j, k1 = j2, n//j2
%o A015634     return n*(n+1)*(n+2)*(n+3)//24-c+j-n # _Chai Wah Wu_, Apr 18 2021
%o A015634 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d)*binomial(d+2, 3))); \\ _Seiichi Manyama_, Jun 12 2021
%o A015634 (PARI) a(n) = binomial(n+3, 4)-sum(k=2, n, a(n\k)); \\ _Seiichi Manyama_, Jun 12 2021
%o A015634 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k/(1-x^k)^4)/(1-x)) \\ _Seiichi Manyama_, Jun 12 2021
%Y A015634 Column k=4 of A177976.
%Y A015634 Partial sums of A117108.
%Y A015634 Cf. A000332, A002088, A013662, A015631, A015650.
%K A015634 nonn
%O A015634 1,2
%A A015634 _Olivier Gérard_