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A049417
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a(n) = isigma(n): sum of infinitary divisors of n.
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26
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1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
Multiplicative: If e = sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = prod_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). Christian G. Bower (bowerc(AT)usa.net) and Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu) May 20 2005.
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LINKS
| Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.
S. R. Finch, Unitarism and infinitarism.
J. O. M. Pedersen, Tables of Aliquot Cycles
R. J. Mathar, Table of n, a(n) for n = 1..7417
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FORMULA
| Let n = product q_i where {q_i} is a set of distinct terms of A050376. Then a(n) = product (q_i+1). - Vladimir Shevelev, Feb 19 2011.
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EXAMPLE
| If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
n=90=2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - Vladimir Shevelev, Feb 19 2011
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MAPLE
| maxpowp := proc(p, n) local f; for f in ifactors(n)[2] do if op(1, f) = p then return op(2, f) ; end if; end do: return 0 ; end proc:
isidiv := proc(d, n) local n2, d2, p, j; if n mod d <> 0 then return false; end if; for p in numtheory[factorset](n) do n2 := maxpowp(p, n) ; n2 := convert(n2, base, 2) ; d2 := maxpowp(p, d) ; d2 := convert(d2, base, 2) ; for j from 1 to nops(d2) do if op(j, n2) = 0 and op(j, d2) <> 0 then return false; end if; end do: end do; return true; end proc:
idivisors := proc(n) local a, d; a := {} ; for d in numtheory[divisors](n) do if isidiv(d, n) then a := a union {d} ; end if; end do: a ; end proc:
A049417 := proc(n) add(d, d=idivisors(n)) ; end proc: # R. J. Mathar, Feb 19 2011
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MATHEMATICA
| Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@
Flatten[Outer[z, Sequence @@ bitty /@
Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]
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CROSSREFS
| Cf. A037445, A004607.
Sequence in context: A034448 A069184 A181549 * A188999 A186644 A125139
Adjacent sequences: A049414 A049415 A049416 * A049418 A049419 A049420
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KEYWORD
| nonn,mult
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AUTHOR
| Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp)
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EXTENSIONS
| More terms from wouter.meeussen(AT)pandora.be, Sep 02, 2001
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