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 A316296 a(n) = Sum_{k=1..n} f(k, n), where f(i, j) is the number of multiples of i greater than j and less than 2*j. 0
 0, 1, 3, 5, 9, 10, 15, 18, 21, 24, 31, 30, 38, 41, 44, 48, 55, 56, 64, 65, 70, 75, 84, 81, 90, 95, 98, 103, 112, 109, 120, 123, 129, 134, 139, 139, 150, 155, 160, 161, 173, 170, 183, 184, 187, 198, 205, 202, 212, 217, 223, 226, 239, 236, 245, 248, 255, 262, 271, 266, 282, 285, 288 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS f(n, m) is the number of multiples of n that are > m and < 2*m. n and m must be both >= 0. By definition, this means that f(n, m) =   0    if n >= 2m;   1    if m < n < 2m; If n <= m, then m = kn + q, where 0 <= q < n. It can be proven that in this case f(n, m) =   k - 1    if q = 0;   k        if q > 0 and (n - q) >= q;   k + 1    if q > 0 and (n - q) < q. Let sd(n) = A006218; then a(n) = sd(2n-1) - sd(n) - (n - 1). Also, a(n) = Sum_{k=n+1..2n-1} (d(k) - 1), where d(k) is number of divisors (A000005). Number of ways the numbers from 1..n divide the numbers from n+1..2n-1, n>=2. - Wesley Ivan Hurt, Feb 08 2022 LINKS FORMULA a(n) = Sum_{k=1..n} Sum_{i=n+1..2n-1} (1-ceiling(i/k)+floor(i/k)). - Wesley Ivan Hurt, Feb 08 2022 EXAMPLE For n = 7, a(7) = f(1,7) + f(2,7) + f(3,7) + f(4,7) + f(5,7) + f(6,7) + f(7,7) = 6 + 3 + 2 + 2 + 1 + 1 = 15. PROG (JavaScript) function f(n, m){     var count = 0;     for(var i=m+1; i<2*m; i++){         if(i%n === 0) count++;     }     return count; } function sf(n){     var sum = 0;     for(var i=1; i<=n; i++){         sum += f(i, n);     }     return sum; } (PARI) a(n) = n + sum(m = 1, n, (floor((n<<1 - 1) / m) - ceil((n + 1) / m))) \\ David A. Corneth, Jun 29 2018 CROSSREFS Sequence in context: A345916 A175468 A286065 * A344293 A063038 A236309 Adjacent sequences:  A316293 A316294 A316295 * A316297 A316298 A316299 KEYWORD nonn,easy,changed AUTHOR Andrea La Rosa, Jun 29 2018 STATUS approved

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Last modified July 2 07:33 EDT 2022. Contains 354985 sequences. (Running on oeis4.)