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A372299
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Primitive infinitary abundant numbers (definition 2): infinitary abundant numbers (A129656) having no proper infinitary divisors that are infinitary abundant numbers.
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3
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24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 642, 654, 660, 678, 726, 756, 762, 780, 786, 822, 834, 894, 906, 924, 942, 945, 960, 978, 990
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OFFSET
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1,1
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LINKS
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EXAMPLE
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24 is a term since it is an infinitary abundant number and none of its proper infinitary divisors, {1, 2, 3, 4, 6, 8, 12}, are infinitary abundant numbers.
The least infinitary abundant number that is not primitive is 120. It has 3 infinitary divisors, 24, 30, and 40, that are also infinitary abundant numbers.
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MATHEMATICA
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f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; iabQ[n_] := isigma[n] > 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], !iabQ[#] &]]; Select[Range[1000], q]
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PROG
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(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); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) > 2*x, idivs(n)) == [];
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CROSSREFS
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KEYWORD
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nonn,easy,new
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AUTHOR
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STATUS
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approved
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