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A049417 a(n) = isigma(n): sum of infinitary divisors of n. 32
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; text; internal format)
OFFSET

1,2

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 and Mitch Harris, May 20 2005

This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = prod {q is in Q_n}(q+1) =  n*prod {q is in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..7417

Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp. 54 (189) (1990) 395-411.

Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

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]

Tomohiro Yamada, Infinitary superperfect numbers, arXiv:1705.10933 [math.NT], 2017.

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

If n is squarefree, then a(n) = A001615(n). - Vladimir Shevelev, Apr 01 2014

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

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

MATHEMATICA

bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]

PROG

(PARI) A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014

(Haskell)

a049417 1 = 1

a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where

   f p e = product $ zipWith div

           (map (subtract 1 . (p ^)) $

                zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)

           (map (subtract 1 . (p ^)) a000079_list)

-- Reinhard Zumkeller, Sep 18 2015

CROSSREFS

Cf. A037445, A004607.

Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).

Cf. A000079, A030308, A027748, A124010.

Sequence in context: A069184 A181549 A241405 * A188999 A186644 A125139

Adjacent sequences:  A049414 A049415 A049416 * A049418 A049419 A049420

KEYWORD

nonn,mult

AUTHOR

Yasutoshi Kohmoto, Dec 11 1999

EXTENSIONS

More terms from Wouter Meeussen, Sep 02 2001

STATUS

approved

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Last modified August 19 01:37 EDT 2017. Contains 290788 sequences.