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.

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

Offset: 1

Author

Frank M Jackson, Jan 10 2016

Keywords

• nonn

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 the primitive by an integer formed from power combinations of the divisors of the primitive (see A379325 and A379713). [Comment corrected by Frank M Jackson, Jan 01 2025]

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A005117, A005153, A265501, A379325, A379713.

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

Programs

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

PARI

\\ see A005153 for is_A005153
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);}
is_A267124(n) = is_A005153(n) && (issquarefree(n) || isp(n)); \\ Michel Marcus, Jun 19 2019. [Name edited for use in A361872 and elsewhere. - M. F. Hasler, Jun 20 2023]

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
def is_A267124(n):
    if is_A005153(n) and is_primitive(n):
        return True # Karl-Heinz Hofmann, Mar 09 2023