cp's OEIS Frontend

A265501 Practical numbers that are squarefree.

%I A265501 #31 Feb 07 2025 00:45:38
%S A265501 1,2,6,30,42,66,78,210,330,390,462,510,546,570,690,714,798,858,870,
%T A265501 930,966,1110,1122,1218,1230,1254,1290,1302,1326,1410,1482,1518,1554,
%U A265501 1590,1722,1770,1794,1806,1830,1914,1974,2010,2046,2130,2190,2226,2262,2310
%N A265501 Practical numbers that are squarefree.
%C A265501 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).
%C A265501 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
%H A265501 Amiram Eldar, <a href="/A265501/b265501.txt">Table of n, a(n) for n = 1..10000</a>
%H A265501 Carl Pomerance, Lola Thompson, and Andreas Weingartner, <a href="https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/175/3/91743/on-integers-n-for-which-x-n-1-has-a-divisor-of-every-degree">On integers n for which X^n-1 has a divisor of every degree</a>, Acta Arithmetica 175 (2016), 225-243; <a href="https://arxiv.org/abs/1511.03357">arXiv preprint</a>, arXiv:1511.03357 [math.NT], 2015.
%H A265501 Eric Saias, <a href="https://doi.org/10.1006/jnth.1997.2057">Entiers à diviseurs denses 1</a>, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
%H A265501 Andreas Weingartner, <a href="https://doi.org/10.1090/mcom/3402">On the constant factor in several related asymptotic estimates</a>, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; <a href="https://arxiv.org/abs/1705.06349">arXiv preprint</a>, arXiv:1705.06349 [math.NT], 2017-2018.
%H A265501 Andreas Weingartner, <a href="https://www.worldscientific.com/doi/10.1142/S1793042120500311">The constant factor in the asymptotic for practical numbers</a>, Int. J. Number Theory, 16 (2020), no. 3, 629-638;  <a href="http://arxiv.org/abs/1906.07819">arXiv preprint</a>, arXiv:1906.07819 [math.NT], 2019.
%F A265501 a(n) = C*n*log(n*log(n)) + O(n), where C = 11.447... (see comments). - _Andreas Weingartner_, Jan 24 2025
%e A265501 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.
%t A265501 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]]]
%o A265501 (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);
%o A265501 for(n=1, 10^4, if(is_pr(n) && issquarefree(n), print1(n, ", "))) \\ _Altug Alkan_, Dec 10 2015
%Y A265501 Cf. A005117, A005153.
%K A265501 nonn
%O A265501 1,2
%A A265501 _Frank M Jackson_, Dec 09 2015