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 A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity. 30
 1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For more information see the Comments in A061020. a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015 If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017 Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017 Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020 REFERENCES P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014. FORMULA a(n) = abs(A061020(n)). a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012 Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015 From Michel Marcus, Nov 05 2017: (Start) a(2^n) = A001045(n+1); a(3^n) = A015518(n+1); a(5^n) = A015531(n+1); a(7^n) = A015552(n+1); a(11^n) = A015592(n+1). (End) a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017 a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018 a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018 Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019 G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019 a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019 a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020 MAPLE a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]): seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017 MATHEMATICA Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *) a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *) f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *) PROG (Haskell) a206369 n = product \$    zipWith h (a027748_row n) (map toInteger \$ a124010_row n) where            h p e = sum \$ take (fromInteger e + 1) \$                          iterate ((* p) . negate) (1 - 2 * (e `mod` 2)) -- Reinhard Zumkeller, Feb 08 2012 (PARI) a(n) = sum(k=1, n, issquare(gcd(n, k))); (PARI) ak(p, e)=my(s=1); for(i=1, e, s=s*p + (-1)^i); s a(n)=my(f=factor(n)); prod(i=1, #f~, ak(f[i, 1], f[i, 2])) \\ Charles R Greathouse IV, Dec 27 2016 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018 (PARI) apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020 CROSSREFS Cf. A061020, A206368. Cf. A027748 row, A124010, A206475 (first differences). Cf. A078429. Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect). Sequence in context: A109746 A286365 A061020 * A152958 A278963 A308085 Adjacent sequences:  A206366 A206367 A206368 * A206370 A206371 A206372 KEYWORD nonn,mult AUTHOR N. J. A. Sloane, Feb 06 2012 STATUS approved

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Last modified May 6 09:45 EDT 2021. Contains 343580 sequences. (Running on oeis4.)