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A067392
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Sum of numbers <= n which have common prime factors with n.
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11
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0, 2, 3, 6, 5, 15, 7, 20, 18, 35, 11, 54, 13, 63, 60, 72, 17, 117, 19, 130, 105, 143, 23, 204, 75, 195, 135, 238, 29, 345, 31, 272, 231, 323, 210, 450, 37, 399, 312, 500, 41, 651, 43, 550, 495, 575, 47, 792, 196, 775, 510, 754, 53, 999, 440, 924, 627, 899, 59
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OFFSET
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1,2
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COMMENTS
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Sum of k <= n such that gcd(n,k) > 1.
V. S. Guba conjectured that for any positive n and prime p, a(n) != a(n+p). - Max Alekseyev, May 08 2024
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LINKS
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FORMULA
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Not multiplicative.
a(p) = p where p is a prime; a(2^k) = 2^(k-1)*(2^(k-1) + 1).
Sum_{k=1..n} a(k) ~ (1/6 - 1/(Pi^2)) * n^3. - Amiram Eldar, Dec 03 2023
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EXAMPLE
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For n=24, a(24) = 2+3+4+6+8+9+10+12+14+15+16+18+20+21+22+24 = 204.
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MATHEMATICA
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a[n_] := Plus@@Select[Range[1, n], GCD[ #, n]>1&]
Join[{0}, Table[n (n + 1) / 2 - n EulerPhi@(n) / 2, {n, 2, 60}]] (* Vincenzo Librandi, Jul 19 2019 *)
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PROG
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(PARI) A067392(n)={a=0; for(i=1, n, if(gcd(i, n)<>1, a=a+i)); a}
(PARI) a(n) = sum(k=1, n, k*(gcd(k, n) != 1)); \\ Michel Marcus, May 08 2018
(PARI) a(n) = if(n == 1, 0, n*(n + 1 - eulerphi(n))/2); \\ Amiram Eldar, Dec 03 2023
(Magma) [0] cat [n*(n+1)/2-n*EulerPhi(n)/2: n in [2..60]]; // Vincenzo Librandi, Jul 19 2019
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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