A351362 Number of ways the numbers from 1..n do not divide the numbers from n..2n-1.
0, 1, 4, 8, 14, 22, 32, 42, 57, 72, 88, 108, 129, 151, 177, 203, 232, 262, 295, 329, 367, 405, 443, 487, 532, 577, 627, 675, 727, 783, 839, 895, 956, 1018, 1082, 1148, 1217, 1285, 1357, 1431, 1506, 1586, 1664, 1746, 1832, 1914, 2002, 2092, 2186, 2277, 2374, 2472, 2568, 2672
Offset: 1
Keywords
Examples
a(5) = 14; there are 14 ways that the numbers 1..5 do not divide the numbers 5..9. 2 does not divide 5,7,9 (3 ways) + 3 does not divide 5,7,8 (3 ways) + 4 does not divide 5,6,7,9 (4 ways) + 5 does not divide 6,7,8,9 (4 ways) = 14 ways.
Programs
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Python
def A351362(n): return 1 if n == 2 else n*n-1-sum((2*n-1)//k for k in range(2,2*n-1))+sum((n-1)//k for k in range(2,n-1)) # Chai Wah Wu, Feb 08 2022
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Python
from math import isqrt def A351362(n): return ((t:=isqrt(m:=(n<<1)-1))+(s:=isqrt(r:=n-1)))*(t-s)+(sum(r//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1)+n*(n+1)-1 # Chai Wah Wu, Oct 23 2023
Formula
a(n) = Sum_{k=1..n} Sum_{i=n..2n-1} sign(i mod k).