A265501 Practical numbers that are squarefree.

1, 2, 6, 30, 42, 66, 78, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1326, 1410, 1482, 1518, 1554, 1590, 1722, 1770, 1794, 1806, 1830, 1914, 1974, 2010, 2046, 2130, 2190, 2226, 2262, 2310

Offset: 1

Author

Frank M Jackson, Dec 09 2015

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A005117, A005153.

Programs

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]]]

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

Formula

a(n) = C*n*log(n*log(n)) + O(n), where C = 11.447... (see comments). - Andreas Weingartner, Jan 24 2025