OFFSET
1,2
COMMENTS
All practical numbers greater than 2 are either equivalent to 0 (mod 4) or 0 (mod 6), but 4 is not squarefree so a(n) for n > 2 must always be equivalent to 0 (mod 6).
Let N(x) be the number of terms less than x. Saias (1997) showed that N(x) has order of magnitude x/log(x). We have N(x) = c*x/log(x) + O(x/(log(x))^2), where c=0.087354... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 11.447... Although this result is not in the literature, it follows from the methods in Pomerance, Thompson & Weingartner (2016), Weingartner (2019), Weingartner (2020). - Andreas Weingartner, Jan 24 2025
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance, Lola Thompson, and Andreas Weingartner, On integers n for which X^n-1 has a divisor of every degree, Acta Arithmetica 175 (2016), 225-243; arXiv preprint, arXiv:1511.03357 [math.NT], 2015.
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.
Andreas Weingartner, The constant factor in the asymptotic for practical numbers, Int. J. Number Theory, 16 (2020), no. 3, 629-638; arXiv preprint, arXiv:1906.07819 [math.NT], 2019.
FORMULA
a(n) = C*n*log(n*log(n)) + O(n), where C = 11.447... (see comments). - Andreas Weingartner, Jan 24 2025
EXAMPLE
a(4) = 30 = 2*3*5. It is squarefree and has 7 aliquot divisors: (1, 2, 3, 5, 6, 10, 15). All positive integers less than 30 can be represented by sums of distinct members of this set so 30 is therefore a practical number. It is the fourth such occurrence.
MATHEMATICA
practicalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1||(n > 1 && OddQ[n])||(n > 2 && Mod[n, 4] != 0 && Mod[n, 6] != 0), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod * p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Select[practicalQ][Select[SquareFreeQ][Range[2500]]]
PROG
(PARI) is_pr(n)=bittest(n, 0) && return(n==1); my(P=1); n && !for(i=2, #n=factor(n)~, n[1, i]>1+(P*=sigma(n[1, i-1]^n[2, i-1])) && return);
for(n=1, 10^4, if(is_pr(n) && issquarefree(n), print1(n, ", "))) \\ Altug Alkan, Dec 10 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Dec 09 2015
STATUS
approved