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

20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752, 1848964, 8353792, 8378368, 8382464, 35021696, 45335936, 120888092, 134193152, 775397948, 1845991216, 2146926592, 2146992128, 3381872252

Offset: 1

Author

Amiram Eldar, Apr 06 2023

Keywords

• nonn

Comments

The abundancy index of an integer k is sigma(k)/k, where sigma is the sum-of-divisors function (A000203).

Terms k of A071395 such that sigma(k)/k < sigma(m)/m for all smaller terms m < k of A071395.

Examples

			The abundancy indices of the first terms are 21/10 > 72/35 > 45/22 > 105/52 > 465/232 > 651/325 > 945/472 > ... > 2.
		

Crossrefs

Cf. A000203, A071395, A071927.

Other sequences related to records in A071395: A083873, A334419.

Programs

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
(* Returns the abundancy index of n if n is primitive abundant, and 0 otherwise: *)
abIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; abIndex[1] = 0;
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]

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.
lista(kmax) = {my(ab, abm = 3); for(k = 1, kmax, ab = abindex(k); if(ab > 0 && ab < abm, abm = ab; print1(k, ", "))); }