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A362053 Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.

%I A362053 #8 Apr 07 2023 17:42:41
%S A362053 20,70,88,104,464,650,1888,1952,4030,5830,8925,17816,32128,77744,
%T A362053 91388,128768,130304,442365,521728,522752,1848964,8353792,8378368,
%U A362053 8382464,35021696,45335936,120888092,134193152,775397948,1845991216,2146926592,2146992128,3381872252
%N A362053 Primitive abundant numbers k (A071395) whose abundancy index sigma(k)/k has a record low value.
%C A362053 The abundancy index of an integer k is sigma(k)/k, where sigma is the sum-of-divisors function (A000203).
%C A362053 Terms k of A071395 such that sigma(k)/k < sigma(m)/m for all smaller terms m < k of A071395.
%e A362053 The abundancy indices of the first terms are 21/10 > 72/35 > 45/22 > 105/52 > 465/232 > 651/325 > 945/472 > ... > 2.
%t A362053 f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
%t A362053 (* Returns the abundancy index of n if n is primitive abundant, and 0 otherwise: *)
%t A362053 abIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; abIndex[1] = 0;
%t A362053 seq[kmax_] := Module[{s = {}, ab, abm = 3}, Do[If[0 < (ab = abIndex[k]) < abm, abm = ab; AppendTo[s, k]], {k, 1,  kmax}]; s]; seq[10^6]
%o A362053 (PARI) abindex(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2, r, 0);} \\ Returns the abundancy index of n if n is primitive abundant, and 0 otherwise.
%o A362053 lista(kmax) = {my(ab, abm = 3); for(k = 1, kmax, ab = abindex(k); if(ab > 0 && ab < abm, abm = ab; print1(k, ", "))); }
%Y A362053 Cf. A000203, A071395, A071927.
%Y A362053 Other sequences related to records in A071395: A083873, A334419.
%K A362053 nonn
%O A362053 1,1
%A A362053 _Amiram Eldar_, Apr 06 2023