|
%I #24 Oct 21 2019 11:24:37
%S 1,2,6,30,42,66,78,210,330,390,462,510,546,570,690,714,798,858,870,
%T 930,966,1110,1122,1218,1230,1254,1290,1302,1326,1410,1482,1518,1554,
%U 1590,1722,1770,1794,1806,1830,1914,1974,2010,2046,2130,2190,2226,2262,2310
%N Practical numbers that are squarefree.
%C 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).
%H Amiram Eldar, <a href="/A265501/b265501.txt">Table of n, a(n) for n = 1..10000</a>
%e 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 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 (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 for(n=1, 10^4, if(is_pr(n) && issquarefree(n), print1(n, ", "))) \\ _Altug Alkan_, Dec 10 2015
%Y Cf. A005117, A005153.
%K nonn
%O 1,2
%A _Frank M Jackson_, Dec 09 2015
|
