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

%I A077609 #43 Feb 16 2025 08:32:48
%S A077609 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,
%T A077609 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,
%U A077609 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
%N A077609 Triangle in which n-th row lists infinitary divisors of n.
%C A077609 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
%C A077609 The concept of infinitary divisors was introduced by Cohen (1990). - _Amiram Eldar_, Mar 09 2024
%H A077609 Reinhard Zumkeller, <a href="/A077609/b077609.txt">Rows n = 1..1000 of table, flattened</a>
%H A077609 Graeme L. Cohen, <a href="https://doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
%H A077609 Graeme L. Cohen and Peter Hagis, Jr., <a href="http://dx.doi.org/10.1155/S0161171293000456">Arithmetic functions associated with the infinitary divisors of an integer</a>, Internat. J. Math. Math. Sci. 16 (2) (1993) 373-384.
%H A077609 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>.
%e A077609 The first few rows are:
%e A077609   1;
%e A077609   1, 2;
%e A077609   1, 3;
%e A077609   1, 4;
%e A077609   1, 5;
%e A077609   1, 2, 3, 6;
%e A077609   1, 7;
%e A077609   1, 2, 4, 8;
%e A077609   1, 9;
%e A077609   1, 2, 5, 10;
%e A077609   1, 11;
%e A077609   1, 3, 4, 12;
%e A077609   1, 13;
%e A077609   1, 2, 7, 14;
%e A077609   1, 3, 5, 15;
%e A077609   1, 16;
%e A077609   1, 17;
%p A077609 # see the function idivisors() in A049417. # _R. J. Mathar_, Oct 05 2017
%t A077609 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 *)
%o A077609 (Haskell)
%o A077609 import Data.List ((\\))
%o A077609 a077609 n k = a077609_row n !! (k-1)
%o A077609 a077609_row n = filter
%o A077609    (\d -> d == 1 || null (a213925_row d \\ a213925_row n)) $ a027750_row n
%o A077609 a077609_tabf = map a077609_row [1..]
%o A077609 -- _Reinhard Zumkeller_, Jul 10 2013
%o A077609 (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);}
%o A077609 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
%Y A077609 Cf. A027750, A037445 (row lengths), A049417 (row sums).
%Y A077609 Cf. A222266.
%K A077609 nonn,tabf
%O A077609 1,3
%A A077609 _Eric W. Weisstein_, Nov 11 2002