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

1, 2, 6, 4, 20, 12, 42, 8, 36, 40, 110, 24, 156, 84, 120, 16, 272, 72, 342, 80, 252, 220, 506, 48, 400, 312, 216, 168, 812, 240, 930, 32, 660, 544, 840, 144, 1332, 684, 936, 160, 1640, 504, 1806, 440, 720, 1012, 2162, 96, 1764, 800, 1632, 624, 2756, 432, 2200, 336

Offset: 1

Author

Reinhard Zumkeller, Jan 24 2003

Keywords

• nonn
• mult

Comments

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003958, A008683, A027746, A065488, A068468.

Programs

Haskell

a079579 1 = 1
a079579 n = product $ zipWith (*) pfs $ map (subtract 1) pfs
   where pfs = a027746_row n
-- Reinhard Zumkeller, Jan 05 2012

Mathematica

f[p_, e_] := ((p - 1)*p)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Oct 23 2022 *)

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

Formula

a(n) <= n^2.

a(n) = n iff n = 2^k.

a(n) = n*A003958(n).

Multiplicative sequence with a(p^e) = p^e*(p-1)^e for prime p. - Jaroslav Krizek, Nov 01 2009

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

From Amiram Eldar, Oct 23 2022: (Start)

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... .

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2-p-1)) (A065488). (End)