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 A006579 a(n) = Sum_{k=1..n-1} gcd(n,k). (Formerly M0941) 14
 0, 1, 2, 4, 4, 9, 6, 12, 12, 17, 10, 28, 12, 25, 30, 32, 16, 45, 18, 52, 44, 41, 22, 76, 40, 49, 54, 76, 28, 105, 30, 80, 72, 65, 82, 132, 36, 73, 86, 140, 40, 153, 42, 124, 144, 89, 46, 192, 84, 145, 114, 148, 52, 189, 134, 204, 128, 113, 58, 300, 60, 121, 210, 192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS This sequence for a(n) also arises in the following context. If f(x) is a monic univariate polynomial of degree d>1 over Zn (= Z/nZ, the ring of integers modulo n), and we let X be the number of distinct roots of f(x) in Zn taken over all n^d choices for f(x), then the variance Var[X] = a(n)/n and the expected value E[X] = 1. - Michael Monagan, Sep 11 2015 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from T. D. Noe) Marc Le Brun, Email to N. J. A. Sloane, Jul 1991. Michael Monagan and Baris Tuncer, Some results on counting roots of polynomials and the Sylvester resultant, arXiv:1609.08712 [math.CO], 2016. FORMULA a(p) = p-1 for a prime p. a(n) = A018804(n)-n = Sum_{ d divides n } (d-1)*phi(n/d). - Vladeta Jovovic, May 04 2002 a(n+2) = Sum_{k=0..n} gcd(n-k+1, k+1) = -Sum_{k=0..4n+2} gcd(4n-k+3, k+1)*(-1)^k/4. - Paul Barry, May 03 2005 G.f.: Sum_{k>=1} phi(k) * x^(2*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Feb 06 2020 a(p^k) = k(p-1)p^(k-1) for prime p. - Chai Wah Wu, May 15 2022 EXAMPLE a(12) = gcd(12,1) + gcd(12,2) + ... + gcd(12,11) = 1 + 2 + 3 + 4 + 1 + 6 + 1 + 4 + 3 + 2 + 1 = 28. MAPLE a:= n-> add(igcd(n, k), k=1..n-1): seq(a(n), n=1..64); MATHEMATICA f[n_] := Sum[ GCD[n, k], {k, 1, n - 1}]; Table[ f[n], {n, 1, 60}] f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Apr 26 2023 *) PROG (PARI) A006579(n) = sum(k=1, n-1, gcd(n, k)) \\ Michael B. Porter, Feb 23 2010 (Python) from math import prod from sympy import factorint def A006579(n): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) - n # Chai Wah Wu, May 15 2022 CROSSREFS Antidiagonal sums of array A003989. Cf. A018804. Sequence in context: A096189 A010464 A187209 * A227906 A366974 A346004 Adjacent sequences: A006576 A006577 A006578 * A006580 A006581 A006582 KEYWORD nonn AUTHOR Marc LeBrun EXTENSIONS More terms from Robert G. Wilson v, May 04 2002 Corrected by Ron Lalonde (ronronronlalonde(AT)hotmail.com), Oct 24 2002 STATUS approved

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Last modified February 28 03:01 EST 2024. Contains 370379 sequences. (Running on oeis4.)