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A077609
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Triangle in which n-th row lists infinitary divisors of n.
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61
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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EXAMPLE
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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;
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(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..]
(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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