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A265501 Practical numbers that are squarefree. 7
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 (list; graph; refs; listen; history; text; internal format)
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: A330326 A006954 A286652 * A090801 A166062 A100194

Adjacent sequences:  A265498 A265499 A265500 * A265502 A265503 A265504

KEYWORD

nonn

AUTHOR

Frank M Jackson, Dec 09 2015

STATUS

approved

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Last modified July 26 09:29 EDT 2021. Contains 346294 sequences. (Running on oeis4.)