This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A037445 Number of infinitary divisors (or i-divisors) of n. 59
 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 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 For n = 8, n = 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. For n = 90, n = 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) do         a := a*2^wt(p) ;     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} divisors[Infinity][n_] := Sort[Flatten[Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^y[m])]]] Length /@ divisors[Infinity] /@ Range - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 29 2005 a = 1; a[n_] := Times @@ Flatten[ 2^DigitCount[#, 2, 1]&  /@ FactorInteger[n][[All, 2]] ]; Table[a[n], {n, 1, 105}] (* Jean-François 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 * A318307 A286324 A318472 Adjacent sequences:  A037442 A037443 A037444 * A037446 A037447 A037448 KEYWORD nonn,nice,easy,mult AUTHOR EXTENSIONS Corrected and extended by Naohiro Nomoto, Jun 21 2001 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 18 19:59 EST 2019. Contains 329288 sequences. (Running on oeis4.)