A335030 -id:A335030 - cp's OEIS frontend

A335029 Numbers that are not practical (A237287) and have more divisors than any smaller number that is not practical.

3, 9, 10, 44, 70, 225, 315, 770, 1575, 2835, 3465, 10010, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815, 218243025, 509233725, 654729075, 1003917915

Offset: 1

Amiram Eldar, May 20 2020

nonn

The corresponding numbers of divisors are 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 192, 216, 240, 256, 288, 320, 384, 432, 480, 512, ...
Of the first 39 terms, 34 terms are also in A038547.
None of the terms are highly composite (A002182) since all the highly composite numbers are practical numbers (A005153).

			The first 5 numbers that are not practical are 3, 5, 7, 9, 10. Their numbers of divisors are 2, 2, 2, 3, 4. The record numbers of divisors are 2, 3 and 4 which occur at 3, 9 and 10.

Cf. A002182, A005153, A237287, A275239, A335008, A335030.

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

A335029 Numbers that are not practical (A237287) and have more divisors than any smaller number that is not practical.

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