A362052 Practical numbers (A005153) that are abundant and have a record low value of abundancy index.

12, 18, 20, 88, 104, 464, 1888, 1952, 29056, 29312, 29824, 30592, 30848, 32128, 127744, 128768, 130304, 521728, 522752, 8341504, 8353792, 8378368, 8382464, 134029312, 134045696, 134094848, 134193152, 2146926592, 2146992128, 8586723328, 8587902976, 8589082624

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).

All the perfect numbers (A000396) are practical, and their abundancy index is 2.

If k is a deficient practical number, then sigma(k) = 2*k - 1 (i.e., k is an almost-perfect number, and the only known such numbers are the powers of 2, A000079), so the abundancy index of these numbers approaches to the limit 2 from below.

All the terms are either of the form 2^m*p, where p < 2^(m+1) - 1 is a prime, or of the form 2^m*p^2, where p = 2^(m+1) - 1 is a prime.

This sequence is infinite since the abundancy index of practical numbers can be arbitrarily close to 2 from above: if k = 2^m*p, and p < 2^(m+1) - 1 then k is practical, and its abundancy index is (2-1/2^m)*(1+1/p) < 2 + 2/p. Therefore, for all eps > 0, taking a prime p and m such that 2/eps < p < 2^(m+1) - 1 will yield a practical number k = 2^m*p with 2 < sigma(k)/k < 2 + eps.

Examples

			The abundancy indices of the first terms are 7/3 > 13/6 > 21/10 > 45/22 > 105/52 > 465/232 > 945/472 > ... > 2.
		

Links

• Wikipedia, Practical number.

Crossrefs

Subsequence of A005101 and A005153.

Cf. A000079, A000203, A000396, A071927, A335029, A335030.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@ fct]), _?(# > 1 &)] == {};
seq = {}; rm = 3; Do[fct = FactorInteger[n]; r = Times @@ (((First /@ fct)^(1 + Last /@ fct) - 1)/(First /@ fct - 1))/n; If[2 < r < rm && pracQ[fct], rm = r; AppendTo[seq, n]], {n, 3, 10^6}]; seq

PARI

lista(kmax) = {my(f, r, rm = 3, prd, prac); forstep(k = 2, kmax, 2, f = factor(k); r = sigma(f, -1); if(r > 2 && r < rm, prd = 1; prac = 1; for(i=2, #f~, prd *= sigma(f[i-1, 1]^f[i-1, 2]); if(f[i, 1] > 1 + prd, prac = 0; break)); if(prac, rm = r; print1(k, ", ")))); }