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A291784
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a(n) = (psi(n) + phi(n))/2.
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15
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1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 14, 13, 15, 16, 16, 17, 21, 19, 22, 22, 23, 23, 28, 25, 27, 27, 30, 29, 40, 31, 32, 34, 35, 36, 42, 37, 39, 40, 44, 41, 54, 43, 46, 48, 47, 47, 56, 49, 55, 52, 54, 53, 63, 56, 60, 58
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OFFSET
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1,2
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COMMENTS
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This is (A001615 + A000010)/2. It is easy to see that this is always an integer.
If n is a power of a prime (including 1 and primes), then a(n) = n, and in any other case a(n) > n. - M. F. Hasler, Sep 09 2017
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41 (page 96 of 2nd ed., pages 147ff of 3rd ed.).
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LINKS
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N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
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FORMULA
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Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 21/(4*Pi^2) = 0.531936... . - Amiram Eldar, Dec 05 2023
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MAPLE
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f:= proc(n) local P, p;
P:= numtheory:-factorset(n);
n*(mul((p-1)/p, p=P) + mul((p+1)/p, p=P))/2
end proc:
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MATHEMATICA
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psi[n_] := If[n == 1, 1, n*Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := (psi[n] + EulerPhi[n])/2;
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PROG
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(PARI) A291784(n)=(eulerphi(n)+n*sumdivmult(n, d, issquarefree(d)/d))\2 \\ M. F. Hasler, Sep 03 2017
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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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STATUS
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approved
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