A049418 3-i-sigma(n): sum of 3-infinitary divisors of n: if n=Product p(i)^r(i) and d=Product p(i)^s(i), each s(i) has a digit a<=b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-i-divisor of n.

1, 3, 4, 7, 6, 12, 8, 9, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 36, 31, 42, 28, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 54, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 84, 72, 72, 80, 90, 60, 168, 62, 96, 104, 73, 84, 144, 68, 126, 96

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

Cf. A049417 (2-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).

Cf. A000244, A030341, A027748, A124010.

Sequence in context: A073183 A353900 A366903 * A333926 A051378 A366440

Adjacent sequences: A049415 A049416 A049417 * A049419 A049420 A049421

Keyword

nonn,nice,easy,mult

Author

Yasutoshi Kohmoto

Extensions

More terms from Naohiro Nomoto, Sep 10 2001

Status

approved