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A049418
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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.
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7
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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
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OFFSET
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1,2
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COMMENTS
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Multiplicative. If e = sum_{k >= 0} d_k 3^k (base 3 representation), then a(p^e) = prod_{k >= 0} (p^(3^k*{d_k+1}) - 1)/(p^(3^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]
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FORMULA
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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
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EXAMPLE
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Let n = 28 = 2^2*7. Then a(n) = (2^2 + 2 + 1)*(7 + 1) = 56. - Vladimir Shevelev, May 07 2013
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A049417 (2-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
Cf. A000244, A030341, A027748, A124010.
Sequence in context: A073185 A284341 A073183 * A333926 A051378 A254981
Adjacent sequences: A049415 A049416 A049417 * A049419 A049420 A049421
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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Yasutoshi Kohmoto
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EXTENSIONS
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More terms from Naohiro Nomoto, Sep 10 2001
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STATUS
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approved
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