OFFSET
1,5
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(2^e) = e-1 if e>0, a(3^e) = 1, a(p^e) = e+1 if p>3.
Moebius transform is period 12 sequence [ 1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, ...].
G.f.: Sum_{k>0} (x^k - x^(2*k) + x^(4*k) + x^(5*k) + x^(7*k) + x^(8*k) - x^(10*k) + x^(11*k)) / (1 - x^(12*k)). - Michael Somos, Sep 20 2005
a(3*n) = a(n). a(4*n + 2) = 0. - Michael Somos, Nov 16 2011
a(4*n) = A035191(n). - Michael Somos, Mar 19 2015
From Amiram Eldar, Nov 30 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s))*(1 - 1/3^s).
Sum_{k=1..n} a(k) ~ n*log(n)/3 + (2*gamma - 1 + log(3)/2)*n/3, where gamma is Euler's constant (A001620). (End)
EXAMPLE
G.f. = x + x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + x^9 + 2*x^11 + x^12 + 2*x^13 + ...
a(5)=2 because there are two ways of differences: First pe(3)-pe(-2)=(-15)-(-20)=5 and second pe(1)-pe(2)=(1)-(-4)=5, for e=-4.
MAPLE
res:=1; ifac:=op(ifactors(i))[2]; for pfac in ifac do; if pfac[1]=2 then res:=res*(pfac[2]-1); else if pfac[1]<>3 then res:=res*(pfac[2]+1); fi; fi; od; a(i):=res;
MATHEMATICA
a[ n_] := If[ n < 1, 0, If[ Divisible[n, 4], -1, 1] Sum[ KroneckerSymbol[ -3, d] (-1)^Quotient[ d, 3], {d, Divisors@n}]]; (* Michael Somos, Mar 19 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, if( n%2==0, (valuation(n, 2) -1) * a(n / 2^valuation(n, 2)), if( n%3==0, a(n / 3^valuation(n, 3)), numdiv(n)))) }; /* Michael Somos, Sep 20 2005 */
(PARI) {a(n) = if( n<1, 0, (-1)^(n%4 == 0) * sumdiv( n, d, (-1)^(d\3) * kronecker( -3, d)))}; /* Michael Somos, Nov 16 2011 */
(Haskell)
a099751 n = product $ zipWith f (a027748_row n) (a124010_row n)
where f 2 e = e - 1; f 3 e = 1; f _ e = e + 1
-- Reinhard Zumkeller, Mar 20 2015
CROSSREFS
KEYWORD
mult,easy,nonn
AUTHOR
Volker Schmitt (clamsi(AT)gmx.net), Nov 10 2004
STATUS
approved