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

1,2

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A005117, A005153.

Sequence in context: A006954 A360681 A286652 * A090801 A166062 A100194

Adjacent sequences: A265498 A265499 A265500 * A265502 A265503 A265504

Keyword

nonn

Author

Frank M Jackson, Dec 09 2015

Status

approved