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A037445 Number of infinitary divisors (or i-divisors) of n. 44
1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 8, 2, 8, 4, 4, 4, 4, 4, 8, 2, 4, 4, 4, 2, 8, 2, 8, 8 (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.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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]

Eric Weisstein's World of Mathematics, Infinitary Divisor

Index entries for sequences computed from exponents in factorization of n

FORMULA

Multiplicative with a(p^e) = 2^A000120(e). - David W. Wilson, Sep 01 2001

Let n = q_1*...*q_k, where q_1,...,q_k are different terms of A050376. Then a(n) = 2^k (the number of subsets of a set with k elements is 2^k). - Vladimir Shevelev, Feb 19 2011.

a(n) = product(A000079(A000120(A124010(n,k))): k=1..A001221(n)). - Reinhard Zumkeller, Mar 19 2013

From Antti Karttunen, May 28 2017: (Start)

a(n) = A286575(A156552(n)). [Because multiplicative with a(p^e) = A001316(e).]

a(n) = 2^A064547(n). (End)

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) = 4.

n=90=2*5*9, where 2,5,9 are in A050376; so a(90)=2^3=8.

MAPLE

A037445 := proc(n)

    local a, p;

    a := 1 ;

    for p in ifactors(n)[2] do

        a := a*2^wt(p[2]) ;

    end do:

    a ;

end proc: # R. J. Mathar, May 16 2016

MATHEMATICA

Table[Length@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@

Flatten[Outer[z, Sequence @@ bitty /@

Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 240}]

bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]

y[n_] := Select[Range[0, n], BitOr[n, # ] == n & ] divisors[Infinity][1] := {1} divisors[Infinity][n_] := Sort[Flatten[Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^y[m])]]] Length /@ divisors[Infinity] /@ Range[105] - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 29 2005

a[1] = 1; a[n_] := Times @@ Flatten[ 2^DigitCount[#, 2, 1]&  /@ FactorInteger[n][[All, 2]] ]; Table[a[n], {n, 1, 105}] (* Jean-Fran├žois Alcover, Aug 19 2013, after Reinhard Zumkeller *)

PROG

(PARI) A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ Andrew Lelechenko, May 10 2014

(Haskell)

a037445 = product . map (a000079 . a000120) . a124010_row

-- Reinhard Zumkeller, Mar 19 2013

(Scheme) (define (A037445 n) (if (= 1 n) n (* (A001316 (A067029 n)) (A037445 (A028234 n))))) ;; Antti Karttunen, May 28 2017

(Python)

from sympy import factorint

def wt(n): return bin(n).count("1")

def a(n):

    f=factorint(n)

    return 2**sum([wt(f[i]) for i in f]) # Indranil Ghosh, May 30 2017

CROSSREFS

Cf. A000120, A001316, A004607, A007358, A007357, A038148, A049417, A064547, A074848, A124010, A156552, A286575.

Sequence in context: A084718 A154851 A281854 * A286324 A186643 A286575

Adjacent sequences:  A037442 A037443 A037444 * A037446 A037447 A037448

KEYWORD

nonn,nice,easy,mult

AUTHOR

Yasutoshi Kohmoto

EXTENSIONS

Corrected and extended by Naohiro Nomoto, Jun 21 2001

STATUS

approved

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Last modified August 19 06:42 EDT 2017. Contains 290794 sequences.