|
%I
%S 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,
%T 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,
%U 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
%N Number of infinitary divisors (or i-divisors) of n.
%C 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.
%H Reinhard Zumkeller, <a href="/A037445/b037445.txt">Table of n, a(n) for n = 1..10000</a>
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Unitarism and infinitarism</a>.
%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>
%F Multiplicative with a(p^e) = 2^A000120(e). - David W. Wilson, Sep 01, 2001
%F 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.
%F a(n) = product(A000079(A000120(A124010(n,k))): k=1..A0001221(n)). - _Reinhard Zumkeller_, Mar 19 2013
%e 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.
%e n=90=2*5*9, where 2,5,9 are in A050376; so a(90)=2^3=8.
%t Table[Length@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@
%t Flatten[Outer[z, Sequence @@ bitty /@
%t Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 240}]
%t bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]
%t 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
%o (Haskell)
%o a037445 = product . map (a000079 . a000120) . a124010_row
%o -- _Reinhard Zumkeller_, Mar 19 2013
%Y Cf. A007358, A007357, A038148, A049417, A004607.
%K nonn,nice,easy,mult
%O 1,2
%A Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp)
%E Corrected and extended by _Naohiro Nomoto_, Jun 21 2001
|
