cp's OEIS Frontend

A351355 Number of ways the numbers from 1..n do not divide numbers from n+1..2n.

%I A351355 #21 Aug 26 2025 12:41:25
%S A351355 0,1,3,8,13,21,31,42,55,71,87,107,128,150,174,203,231,260,294,328,364,
%T A351355 404,442,486,530,576,624,674,726,780,838,895,953,1017,1079,1146,1216,
%U A351355 1284,1354,1430,1505,1583,1663,1745,1827,1913,2001,2091,2184,2275,2371,2471,2567,2669,2773
%N A351355 Number of ways the numbers from 1..n do not divide numbers from n+1..2n.
%H A351355 Robert Israel, <a href="/A351355/b351355.txt">Table of n, a(n) for n = 1..10000</a>
%F A351355 a(n) = Sum_{k=1..n} Sum_{i=n+1..2n} sign(i mod k).
%F A351355 a(n) = n*(n+1) + A006218(n) - A006218(2n). - _Chai Wah Wu_, Feb 08 2022
%e A351355 a(5) = 13; there are 13 ways the numbers from 1..5 do not divide the numbers from 6..10. 2 does not divide 7,9 (2 ways) + 3 does not divide 7,8,10 (3 ways) + 4 does not divide 6,7,9,10 (4 ways) + 5 does not divide 6,7,8,9 (4 ways) = 13 ways.
%p A351355 f:= proc(n) local i; n^2 - add(floor(2*n/i) - floor(n/i),i=1..n) end proc:
%p A351355 map(f, [$1..100]); # _Robert Israel_, Aug 26 2025
%o A351355 (Python)
%o A351355 def A351355(n): return 0 if n == 1 else n*n-sum(2*n//k for k in range(2,2*n))+sum(n//k for k in range(2,n)) # _Chai Wah Wu_, Feb 08 2022
%o A351355 (Python)
%o A351355 from math import isqrt
%o A351355 def A351355(n): return ((t:=isqrt(m:=n<<1))+(s:=isqrt(n)))*(t-s)+(sum(n//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1)+n*(n+1) # _Chai Wah Wu_, Oct 23 2023
%Y A351355 Cf. A006218, A077024.
%K A351355 nonn,changed
%O A351355 1,3
%A A351355 _Wesley Ivan Hurt_, Feb 08 2022