A372299 Primitive infinitary abundant numbers (definition 2): infinitary abundant numbers (A129656) having no proper infinitary divisors that are infinitary abundant numbers.
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
Offset: 1
Examples
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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
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] -
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)) == [];