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A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

%I A372298 #12 Mar 18 2025 16:20:19
%S A372298 40,56,70,72,88,104,756,924,945,1092,1188,1344,1386,1428,1430,1596,
%T A372298 1638,1760,1870,2002,2016,2080,2090,2142,2176,2210,2394,2432,2470,
%U A372298 2530,2584,2720,2750,2944,2990,3040,3128,3190,3200,3230,3250,3400,3410,3496,3712,3770
%N A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.
%H A372298 Amiram Eldar, <a href="/A372298/b372298.txt">Table of n, a(n) for n = 1..10000</a>
%e A372298 40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
%e A372298 24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.
%t A372298 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}]];
%t A372298 isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
%t A372298 idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
%t A372298 q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]
%o A372298 (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 A372298 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
%o A372298 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)))} ;
%o A372298 is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];
%Y A372298 Subsequence of A129656 and A372299.
%Y A372298 A372300 is a subsequence.
%Y A372298 Cf. A007357, A077609, A129657.
%Y A372298 Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.
%K A372298 nonn,easy
%O A372298 1,1
%A A372298 _Amiram Eldar_, Apr 25 2024