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A079579 Totally multiplicative with p -> (p-1)*p, p prime.

%I A079579 #23 Oct 23 2022 02:54:05
%S A079579 1,2,6,4,20,12,42,8,36,40,110,24,156,84,120,16,272,72,342,80,252,220,
%T A079579 506,48,400,312,216,168,812,240,930,32,660,544,840,144,1332,684,936,
%U A079579 160,1640,504,1806,440,720,1012,2162,96,1764,800,1632,624,2756,432,2200,336
%N A079579 Totally multiplicative with p -> (p-1)*p, p prime.
%C A079579 The Dirichlet inverse is 1, -2, -6, 0, -20, 12, -42, 0, 0, 40, -110, 0, -156, 84, 120, 0, -272, ..., i.e., the sequence defined by mu(n)*a(n). - _R. J. Mathar_, Dec 20 2011
%H A079579 Reinhard Zumkeller, <a href="/A079579/b079579.txt">Table of n, a(n) for n = 1..10000</a>
%F A079579 a(n) <= n^2.
%F A079579 a(n) = n iff n = 2^k.
%F A079579 a(n) = n*A003958(n).
%F A079579 Multiplicative sequence with a(p^e) = p^e*(p-1)^e for prime p. - _Jaroslav Krizek_, Nov 01 2009
%F A079579 Dirichlet g.f.: sum_{n>=1} a(n)/n^s = Product_{primes p} 1/(1+p^(1-s)-p^(2-s)). - _R. J. Mathar_, Dec 20 2011
%F A079579 From _Amiram Eldar_, Oct 23 2022: (Start)
%F A079579 Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(6)/(3*zeta(2)*zeta(3)) = 2*Pi^4/(945*zeta(3)) = A068468 / 3 = 0.171503... .
%F A079579 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2-p-1)) (A065488). (End)
%t A079579 f[p_, e_] := ((p - 1)*p)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* _Amiram Eldar_, Oct 23 2022 *)
%o A079579 (Haskell)
%o A079579 a079579 1 = 1
%o A079579 a079579 n = product $ zipWith (*) pfs $ map (subtract 1) pfs
%o A079579    where pfs = a027746_row n
%o A079579 -- _Reinhard Zumkeller_, Jan 05 2012
%o A079579 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]*(f[i,1]-1))^f[i,2]); } \\ _Amiram Eldar_, Oct 23 2022
%Y A079579 Cf. A003958, A008683, A027746, A065488, A068468.
%K A079579 nonn,mult
%O A079579 1,2
%A A079579 _Reinhard Zumkeller_, Jan 24 2003