A077609 Triangle in which n-th row lists infinitary divisors of n.

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 4, 7, 28, 1, 29, 1

Offset: 1

Author

Eric W. Weisstein, Nov 11 2002

Keywords

• nonn
• tabf

Comments

The first difference from the triangle A222266 (bi-unitary divisors of n) is in row n = 16; indeed, the 16th row of A222266 is (1, 2, 8, 16) while the 16th of this sequence here is (1, 16). - Bernard Schott, Mar 10 2023

The concept of infinitary divisors was introduced by Cohen (1990). - Amiram Eldar, Mar 09 2024

Examples

			The first few rows are:
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;
  1, 13;
  1, 2, 7, 14;
  1, 3, 5, 15;
  1, 16;
  1, 17;
		

Links

• Reinhard Zumkeller, Rows n = 1..1000 of table, flattened
• Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
• Graeme L. Cohen and Peter Hagis, Jr., Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (2) (1993) 373-384.
• Eric Weisstein's World of Mathematics, Infinitary Divisor.

Crossrefs

Cf. A027750, A037445 (row lengths), A049417 (row sums).

Cf. A222266.

Programs

Haskell

import Data.List ((\\))
a077609 n k = a077609_row n !! (k-1)
a077609_row n = filter
   (\d -> d == 1 || null (a213925_row d \\ a213925_row n)) $ a027750_row n
a077609_tabf = map a077609_row [1..]
-- Reinhard Zumkeller, Jul 10 2013

Maple

# see the function idivisors() in A049417. # R. J. Mathar, Oct 05 2017

Mathematica

f[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ; Array[f, 30] // Flatten (* Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 29 2005 *) (* edited by Michael De Vlieger, Jun 07 2016 *)

PARI

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k,2]); bde = binary(valuation(d, f[k,1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)););); return (1);}
row(n) = {d = divisors(n); f = factor(n); idiv = []; for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k]));); idiv;} \\ Michel Marcus, Feb 15 2016