A335029 -id:A335029 - cp's OEIS frontend

A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

3, 9, 10, 44, 70, 102, 350, 372, 1608, 3492, 6096, 10380, 44040, 100260, 180240, 425160, 1744560, 2425080, 5509980, 10048080, 23614920, 97639920, 396315360, 900229680, 2519017200, 3113704440, 12870562320, 52307529120

Offset: 1

Amiram Eldar, May 20 2020

None of the terms are superabundant (A004394) since all the superabundant numbers are practical numbers (A005153).
The least term m that is k-abundant (having sigma(m)/m > k) for k = 2, 3, ... is A005101(14) = 70, A068403(896) = 44040, A068404(792087) = 3113704440, ...
What is the least 5-abundant number (A215264) that is not practical?

			The first 5 numbers that are not practical are m = 3, 5, 7, 9, 10. Their abundancy indices sigma(m)/m are 1.333..., 1.2, 1.142..., 1.444..., 1.8. The record values occur at 3, 9 and 10.

Cf. A000203 (sigma), A004394, A005153, A237287, A330244, A335029.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; seq = {}; rm = 1; Do[fct = FactorInteger[n]; r = Times@@((First/@fct^ (1+Last/@ fct)-1)/(First/@fct-1))/n; If[r > rm && !pracQ[fct], rm = r; AppendTo[seq, n]], {n, 3, 10^5}]; seq

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

Original entry on oeis.org

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

Amiram Eldar, Apr 06 2023

nonn

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.

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

Wikipedia, Practical number.

Subsequence of A005101 and A005153.

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

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

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, ", ")))); }

cp's OEIS Frontend

A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

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