

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.


8



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.
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a primitive practical number. This is a tighter conjecture to the conjecture posed by Hal M. Switkay (see A005153).
Analogous to the {1 union primes} (A008578) and practical numbers (A005153), the sequence of primitive practical numbers with two extra, practical only, terms added, namely 4 and 8, becomes a "complete" (sic) sequence.  Frank M Jackson, Mar 14 2023


LINKS

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided.  N. J. A. Sloane, May 20 2023]


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

isp(n) = {my(f=factor(n)); for (k=1, #f~, if ((f[k, 2] > 1) && is_A005153(n/f[k, 1]), return (0)); ); return (1); }
(Python)
from sympy import factorint
def is_primitive(n): # uses is_A005153: see there, please DO NOT copy code here!
for i in range(0, len(list(factorint(n)))):
if list(factorint(n).values())[i] > 1:
if is_A005153(n//list(factorint(n))[i]): return False
return True
if is_A005153(n) and is_primitive(n):


CROSSREFS

Superset of primorial numbers (A002110) and superset of perfect numbers (A000396).


KEYWORD

nonn


AUTHOR



STATUS

approved



