

A267124


Primitive practical numbers: practical numbers that are squarefree or practical numbers that when divided by any of its prime factors whose factorization exponent is greater than 1 is no longer practical.


4



1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460, 462, 464, 476, 496, 510, 522, 532, 546, 558, 570, 580, 620, 644, 666, 690, 714, 740, 744, 798, 812, 820, 858, 860, 868, 870, 888, 930, 966, 984
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OFFSET

1,2


COMMENTS

If n is a practical number and d is any of its divisors then n*d must be practical. Consequently the sequence of all practical numbers must contain members that are either squarefree (A265501) or when divided by any of its prime factors whose factorization exponent is greater than 1 is no longer practical. Such practical numbers are said to be primitive. The set of all practical numbers can be generated from the set of primitive practical numbers by multiplying these primitives by any of their divisors.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michel Marcus)
Wikipedia, Complete sequence, Practical number, and Squarefree integer


EXAMPLE

a(4)=20=2^2*5. It is a practical number because it has 6 divisors 1, 2, 4, 5, 10, 20 that form a complete sequence. If it is divided by 2 the resultant has 4 divisors 1, 2, 5, 10 that is not a complete sequence.
a(7)=42=2*3*7. It is squarefree and is practical because it has 8 divisors 1, 2, 3, 6, 7, 14, 21, 42 that form a complete sequence.


MATHEMATICA

PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1(n>1&&OddQ[n]), 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]]]; lst=Select[Range[1, 1000], PracticalQ]; lst1=lst; maxfac=PrimePi[Last[Union[Flatten[FactorInteger[lst], 1]]][[1]]]; Do[lst1=Select[lst1, Mod[#, Prime[p]^2]!=0!PracticalQ[#/Prime[p]] &], {p, 1, maxfac}]; lst1


PROG

(PARI) ispract(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, i1]^n[2, i1])) && return); \\ A005153
isp(n) = {my(f=factor(n)); for (k=1, #f~, if ((f[k, 2] > 1) && ispract(n/f[k, 1]), return (0)); ); return (1); }
isok(n) = ispract(n) && (issquarefree(n)  isp(n)); \\ Michel Marcus, Jun 19 2019


CROSSREFS

Cf. A005117, A005153, A265501.
Sequence in context: A032622 A104749 A062281 * A322371 A131441 A035142
Adjacent sequences: A267121 A267122 A267123 * A267125 A267126 A267127


KEYWORD

nonn


AUTHOR

Frank M Jackson, Jan 10 2016


STATUS

approved



