OFFSET
1,2
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. O. M. Pedersen, Tables of Aliquot Cycles, backup on web.archive.org of no more exiting page, as of May 2014
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
FORMULA
Multiplicative with a(p^e) = prod_{k >= 0} (p^(3^k*{d_k+1}) - 1)/(p^(3^k) - 1), where e = sum_{k >= 0} d_k 3^k (base 3 representation). - Christian G. Bower and Mitch Harris, May 20 2005. [Edited by M. F. Hasler, Sep 21 2022]
Denote P_3 = {p^3^k}, k = 0, 1, ..., p runs primes. Then every n has a unique representation of the form n = prod q_i prod (r_j)^2, where q_i, r_j are distinct elements of P_3. Using this representation, we have a(n) = prod (q_i+1)*prod ((r_j)^2+r_j+1). - Vladimir Shevelev, May 07 2013
EXAMPLE
Let n = 28 = 2^2*7. Then a(n) = (2^2 + 2 + 1)*(7 + 1) = 56. - Vladimir Shevelev, May 07 2013
MAPLE
A049418 := proc(n) option remember; local ifa, a, p, e, d, k ; ifa := ifactors(n)[2] ; a := 1 ; if nops(ifa) = 1 then p := op(1, op(1, ifa)) ; e := op(2, op(1, ifa)) ; d := convert(e, base, 3) ; for k from 0 to nops(d)-1 do a := a*(p^((1+op(k+1, d))*3^k)-1)/(p^(3^k)-1) ; end do: else for d in ifa do a := a*procname( op(1, d)^op(2, d)) ; end do: return a; end if; end proc:
seq(A049418(n), n=1..40) ; # R. J. Mathar, Oct 06 2010
MATHEMATICA
A049418[n_] := Module[{ifa = FactorInteger[n], a = 1, p, e, d, k}, If[ Length[ifa] == 1, p = ifa[[1, 1]]; e = ifa[[1, 2]]; d = Reverse[ IntegerDigits[e, 3] ]; For[k = 1, k <= Length[d], k++, a = a*(p^((1 + d[[k]])*3^(k - 1)) - 1)/(p^(3^(k - 1)) - 1)], Do[ a = a*A049418[ d[[1]]^d[[2]] ], {d, ifa}]]; Return[a] ]; A049418[1] = 1; Table[ A049418[n] , {n, 1, 69}] (* Jean-François Alcover, Jan 03 2012, after R. J. Mathar *)
PROG
(Haskell) following Bower and Harris:
a049418 1 = 1
a049418 n = product $ zipWith f (a027748_row n) (a124010_row n) where
f p e = product $ zipWith div
(map (subtract 1 . (p ^)) $
zipWith (*) a000244_list $ map (+ 1) $ a030341_row e)
(map (subtract 1 . (p ^)) a000244_list)
-- Reinhard Zumkeller, Sep 18 2015
(PARI) apply( {A049418(n)=vecprod([prod(k=1, #n=digits(f[2], 3), (f[1]^(3^(#n-k)*(n[k]+1))-1)\(f[1]^3^(#n-k)-1))|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Sep 21 2022
CROSSREFS
KEYWORD
nonn,nice,easy,mult
AUTHOR
EXTENSIONS
More terms from Naohiro Nomoto, Sep 10 2001
STATUS
approved