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 A145378 a(n) = Sum_{d|n} sigma(d) - 2*Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b). 1
 1, 2, 5, 7, 7, 10, 9, 20, 18, 14, 13, 35, 15, 18, 35, 49, 19, 36, 21, 49, 45, 26, 25, 100, 38, 30, 58, 63, 31, 70, 33, 110, 65, 38, 63, 126, 39, 42, 75, 140, 43, 90, 45, 91, 126, 50, 49, 245, 66, 76, 95, 105, 55, 116, 91, 180, 105, 62, 61, 245, 63, 66, 162, 235, 105, 130, 69 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. See g(n). FORMULA Dirichlet g.f. (1-2/2^s+4/4^s)*(zeta(s))^2*zeta(s-1). Dirichlet convolution of [1,-2,0,4,0,0,0..] with A007429. MAPLE with(numtheory); g:=proc(n) local d, c, b, t0, t1, t2, t3; t1:=divisors(n); t0:=add( sigma(d), d in t1); t2:=0; for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od: t3:=0; for d in t1 do if d mod 4 = 0 then t3:=t3+sigma(d/4); fi; od: t0-2*t2+4*t3; end; [seq(g(n), n=1..100)]; # alternative read("transforms") : nmax := 100 : L27 := [seq(i, i=1..nmax) ]; L := [1, -2, 0, 4, seq(0, i=1..nmax)] ; DIRICHLET(L27, L) : MOBIUSi(%) : MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017 MATHEMATICA a[n_] := Sum[DivisorSigma[1, d] - 2 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2] + 4 Boole[Mod[d, 4] == 0] DivisorSigma[1, d/4], {d, Divisors[n]}]; Array[a, 100] (* Jean-François Alcover, Apr 04 2020 *) CROSSREFS Sequence in context: A226213 A199590 A096624 * A069887 A254340 A120303 Adjacent sequences:  A145375 A145376 A145377 * A145379 A145380 A145381 KEYWORD nonn,mult AUTHOR N. J. A. Sloane, Mar 12 2009 STATUS approved

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Last modified June 20 15:21 EDT 2021. Contains 345165 sequences. (Running on oeis4.)