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 A003958 If n = Product p(k)^e(k) then a(n) = Product (p(k)-1)^e(k). 101
 1, 1, 2, 1, 4, 2, 6, 1, 4, 4, 10, 2, 12, 6, 8, 1, 16, 4, 18, 4, 12, 10, 22, 2, 16, 12, 8, 6, 28, 8, 30, 1, 20, 16, 24, 4, 36, 18, 24, 4, 40, 12, 42, 10, 16, 22, 46, 2, 36, 16, 32, 12, 52, 8, 40, 6, 36, 28, 58, 8, 60, 30, 24, 1, 48, 20, 66, 16, 44, 24, 70, 4, 72, 36, 32, 18, 60, 24, 78, 4, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Completely multiplicative. Dirichlet inverse of A097945. - R. J. Mathar, Aug 29 2011 LINKS Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe) Vaclav Kotesovec, Graph - the asymptotic ratio (10^7 terms). Index to divisibility sequences FORMULA Multiplicative with a(p^e) = (p-1)^e. - David W. Wilson, Aug 01 2001 a(n) = A000010(n) iff n is squarefree (see A005117). - Reinhard Zumkeller, Nov 05 2004 a(n) = abs(A125131(n)). - Tom Edgar, May 26 2014 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4 / (315 * zeta(3)) = 1/(2*A082695) = 0.25725505075419... - Vaclav Kotesovec, Jun 14 2020 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(1-s) + p^(-s)). - Ilya Gutkovskiy, Feb 27 2022 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + (p^(1-s) - 2) / (1 - p + p^s)), (with a product that converges for s=2). - Vaclav Kotesovec, Feb 11 2023 MAPLE a:= n-> mul((i[1]-1)^i[2], i=ifactors(n)[2]): seq(a(n), n=1..80); # Alois P. Heinz, Sep 13 2017 MATHEMATICA DirichletInverse[f_][1] = 1/f[1]; DirichletInverse[f_][n_] := DirichletInverse[f][n] = -1/f[1]*Sum[ f[n/d]*DirichletInverse[f][d], {d, Most[ Divisors[n]]}]; muphi[n_] := MoebiusMu[n]*EulerPhi[n]; Table[ DirichletInverse[ muphi][n], {n, 1, 81}] (* Jean-François Alcover, Dec 12 2011, after R. J. Mathar *) a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jun 10 2016 *) PROG (PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-p*X+X))[n]) /* Ralf Stephan */ (Haskell) a003958 1 = 1 a003958 n = product \$ map (subtract 1) \$ a027746_row n -- Reinhard Zumkeller, Apr 09 2012, Mar 02 2012 (Python) from math import prod from sympy import factorint def a(n): return prod((p-1)**e for p, e in factorint(n).items()) print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Feb 27 2022 CROSSREFS Cf. A003959, A168065, A168066, A027746, A006093, A027748, A124010. Sequence in context: A187202 A345046 A125131 * A326140 A082729 A326069 Adjacent sequences: A003955 A003956 A003957 * A003959 A003960 A003961 KEYWORD nonn,mult,nice AUTHOR Marc LeBrun EXTENSIONS Definition reedited (from formula) by Daniel Forgues, Nov 17 2009 STATUS approved

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Last modified February 28 04:16 EST 2024. Contains 370379 sequences. (Running on oeis4.)