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A090027 Number of distinct lines through the origin in 5-dimensional cube of side length n.

%I A090027 #15 Mar 30 2021 21:09:29
%S A090027 0,31,211,961,2851,7471,15541,31471,55651,95821,152041,239791,351331,
%T A090027 517831,723241,1007041,1352041,1821721,2359051,3082921,3904081,
%U A090027 4956901,6151651,7677901,9334261,11445361,13746181,16566691,19644031,23432851,27408331,32333581
%N A090027 Number of distinct lines through the origin in 5-dimensional cube of side length n.
%C A090027 Equivalently, number of lattice points where the GCD of all coordinates = 1.
%F A090027 a(n) = A090030(5, n).
%F A090027 a(n) = (n+1)^5 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - _Chai Wah Wu_, Mar 30 2021
%e A090027 a(2) = 211 because the 211 points with at least one coordinate=2 all make distinct lines and the remaining 31 points and the origin are on those lines.
%t A090027 aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]];lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1;Table[lines[5, k], {k, 0, 40}]
%o A090027 (Python)
%o A090027 from functools import lru_cache
%o A090027 @lru_cache(maxsize=None)
%o A090027 def A090027(n):
%o A090027     if n == 0:
%o A090027         return 0
%o A090027     c, j = 1, 2
%o A090027     k1 = n//j
%o A090027     while k1 > 1:
%o A090027         j2 = n//k1 + 1
%o A090027         c += (j2-j)*A090027(k1)
%o A090027         j, k1 = j2, n//j2
%o A090027     return (n+1)**5-c+31*(j-n-1) # _Chai Wah Wu_, Mar 30 2021
%Y A090027 Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.
%K A090027 nonn
%O A090027 0,2
%A A090027 _Joshua Zucker_, Nov 25 2003