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

1,3

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

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.

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A222266.

Sequence in context: A180184 A330752 A222266 * A077610 A329534 A317746

Adjacent sequences: A077606 A077607 A077608 * A077610 A077611 A077612

Keyword

nonn,tabf

Author

Eric W. Weisstein, Nov 11 2002

Status

approved