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

1,1

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.

Links

Table of n, a(n) for n=1..33.

Example

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

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]

Prog

(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, ", "))); }

Crossrefs

Cf. A000203, A071395, A071927.

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

Sequence in context: A153728 A234367 A071395 * A357921 A335557 A245856

Adjacent sequences: A362050 A362051 A362052 * A362054 A362055 A362056

Keyword

nonn

Author

Amiram Eldar, Apr 06 2023

Status

approved