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A066760
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a(n) = Sum_{1<=k<=n, k is not a divisor of n and k is not coprime to n} k.
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5
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0, 0, 0, 0, 0, 4, 0, 6, 6, 18, 0, 27, 0, 40, 37, 42, 0, 79, 0, 89, 74, 108, 0, 145, 45, 154, 96, 183, 0, 274, 0, 210, 184, 270, 163, 360, 0, 340, 257, 411, 0, 556, 0, 467, 418, 504, 0, 669, 140, 683, 439, 657, 0, 880, 369, 805, 548, 810, 0, 1183, 0, 928, 779, 930, 502
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OFFSET
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1,6
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COMMENTS
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This function can be used to prove no p^k is perfect or multi-perfect.
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LINKS
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FORMULA
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a(n) = 1 + n*(n+1)/2 - sigma(n) - n*phi(n)/2.
a(n) = Sum_{k=1..n} k * (1 - floor(1/gcd(n,k))) * (ceiling(n/k) - floor(n/k)). - Wesley Ivan Hurt, Jan 06 2024
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EXAMPLE
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There are three integers that satisfy this definition for n = 12, namely 8, 9 and 10. These sum to 27, hence a(12) = 27.
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MAPLE
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f:= n -> 1 + n*(n+1)/2 - numtheory:-sigma(n) - n*numtheory:-phi(n)/2;
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MATHEMATICA
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Table[n(n + 1)/2 + 1 - EulerPhi[n] * n/2 - DivisorSigma[1, n], {n, 2, 65}] (* Robert G. Wilson v, Jul 31 2004 *)
Table[Sum[k * Boole[Not[Divisible[n, k]]] * Boole[GCD[n, k] > 1], {k, n - 1}], {n, 65}] (* Alonso del Arte, Nov 02 2014 *)
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PROG
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(Magma) [0] cat [1 + n*(n+1)/2 - SumOfDivisors(n) - n*EulerPhi(n)/2: n in [2..70]]; // Vincenzo Librandi, Nov 03 2014
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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