A265501 -id:A265501 - cp's OEIS frontend

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

Frank M Jackson, Jan 10 2016

nonn

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

			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.

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

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

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

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

\\ 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]

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

A286652 Unitary practical numbers: numbers n such that every 1 <= k <= usigma(n) is a sum of distinct unitary divisors of n.

Original entry on oeis.org

1, 2, 6, 30, 42, 66, 78, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1050, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1326, 1410, 1470, 1482, 1518, 1554, 1590, 1650, 1722, 1770, 1794, 1806, 1830, 1914, 1950, 1974, 2010, 2046, 2130

Offset: 1

Amiram Eldar, May 27 2017

nonn

The unitary version of A005153. The squarefree terms of both sequences are the same, A265501. The nonsquarefree terms of this sequence are in A287173.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005153, A034448, A265501, A287173.

usigma[n_] :=  Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[#, n/#] == 1 &]]; uPracticalQ[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]; r = Sort[p^e]; Do[If[r[[i]] > 1 + usigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[p]}]; ok]]]; Select[ Range[100000], uPracticalQ]

A287173 Unitary practical numbers that are nonsquarefree.

Original entry on oeis.org

1050, 1470, 1650, 1950, 3234, 3822, 8250, 9438, 9750, 11550, 13650, 16170, 17850, 19110, 19950, 21450, 24150, 24990, 25410, 27930, 28050, 30450, 31350, 32550, 33150, 33810, 35490, 37050, 37950, 38850, 42042, 42630, 43050, 44850, 45150, 45570, 47190, 47850

Offset: 1

Amiram Eldar, May 24 2017

nonn

The squarefree terms of both practical numbers (A005153) and unitary practical numbers (A286652) are the same, A265501.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005153, A034448, A265501, A287173, A286652.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; uPracticalQ[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]; r = Sort[p^e]; Do[If[r[[i]] > 1+usigma[prod], ok=False; Break[]]; prod=prod*r[[i]], {i, Length[p]}]; ok]]]; aQ[n_]:=!SquareFreeQ[n]&&uPracticalQ[n];Select[Range[100000], aQ]

A268156 Smallest squarefree term of adjacent squarefree pairs in the sequence of practical numbers (A005153).

Original entry on oeis.org

1, 3486, 41106, 50358, 77142, 102090, 104610, 118734, 119910, 142662, 155298, 159654, 173910, 192210, 193290, 203010, 205062, 212898, 220818, 228018, 232518, 238170, 239946, 241878, 254478, 265278, 266178, 272118, 273378, 303630, 306210, 311178, 323778, 326370, 331890, 335478, 335946, 336102

Offset: 1

Frank M Jackson, Jan 27 2016

nonn

The first occurrence of adjacent squarefree practical number pairs is 1, 2.

The first occurrence of adjacent squarefree practical number triples is 792834, 792858, 792870.

			a(2) = 3486 = 2*3*7*83 and is squarefree. The next practical number is 3498 = 2*3*11*53 and is also squarefree. This is the second such pairing.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Practical number
Wikipedia, Squarefree integer

Cf. A005117, A005153, A265501.

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[1000000], PracticalQ]; lst1={}; Do[If[SquareFreeQ[lst[[n]]]&&SquareFreeQ[lst[[n+1]]], AppendTo[lst1, lst[[n]]]], {n, 1, Length[lst]-1}]; lst1

A380328 2-dense squarefree numbers: Squarefree numbers whose divisors increase by factors of at most 2.

Original entry on oeis.org

1, 2, 6, 30, 42, 66, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1122, 1218, 1230, 1254, 1290, 1302, 1410, 1518, 1554, 1590, 1722, 1770, 1806, 1914, 1974, 2046, 2226, 2310, 2442, 2478, 2562, 2706, 2730, 2814, 2838, 2982, 3066, 3102, 3318, 3486, 3498

Offset: 1

Frank M Jackson, Jan 21 2025

nonn

This sequence is a subsequence of primitive practical numbers (A267124) because the sequence of 2-dense numbers (A174973) is a subsequence of practical numbers (A005153) and all squarefree practical numbers (A265501) are by definition primitive practical numbers.
Similar to and a subsequence of A265501.
Let N(x) be the number of terms less than x. Saias (1997) showed that N(x) has order of magnitude x/log(x). We have N(x) = c*x/log(x) + O(x/(log(x))^2), where c=0.06864... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 14.56... (see Weingartner (2019)). - Andreas Weingartner, Jan 23 2025

			a(5) = 42 and its prime factorization is 2*3*7 and squarefree. Also the proper divisors are 1, 2, 3, 6, 7, 21, 42 they are 2-dense and therefore 42 is practical as well as being primitive practical.

Frank M Jackson, Table of n, a(n) for n = 1..10000
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.

Intersection of A005117 and A174973.
Subsequence of A267124 and of A265501.
Cf. A005153.

filter:= proc(n) local D,i;
  if not numtheory:-issqrfree(n) then return false fi;
  D:= sort(convert(numtheory:-divisors(n),list));
  andmap(i -> D[i+1]<=2*D[i],[$1..nops(D)-1])
end proc:
select(filter, [1,seq(i,i=2..5000,4)]); # Robert Israel, Jan 23 2025

Dens2DivQ[n_] := Module[{lst=Divisors[n], m, ok}, If[n==1, Return[True]]; Do[ok=False; If[lst[[m+1]]/lst[[m]]>2, Break[]]; ok=True, {m, 1, Length[lst]-1}]; ok]; Select[Range[10000], SquareFreeQ[#]&&Dens2DivQ[#]&]

a(n) = C*n*log(n*log(n)) + O(n), where C = 14.56... (see comments). - Andreas Weingartner, Jan 23 2025

A267422 Largest prime factor of the largest squarefree practical number comprising n prime factors.

Original entry on oeis.org

2, 3, 13, 167, 28219, 796481281, 634382430983400959, 402441068740409482305343048128921493, 161958813808922990180784918278423278413890512706478208244331277280870341

Offset: 1

Frank M Jackson, Jan 14 2016

nonn

The prime factors of the largest squarefree practical number with n prime factors are the first n members of a(n). The largest squarefree practical number with 3 prime factors is 78 = 2*3*13 and the largest squarefree practical number with 5 prime factors is 367580694 = 2*3*13*167*28219, etc.

Because all primorial numbers (A002110) are practical, the prime factors of the smallest squarefree practical number with n prime factors are the first n members of the primes. Hence the smallest squarefree practical number with n prime factors is A002110(n). - Frank M Jackson, May 29 2023

			a(3) = 13 because there are only 4 squarefree practical numbers with 3 prime factors, namely 2*3*5 = 30, 2*3*7 = 42, 2*3*11 = 66 and 2*3*13 = 78. So 78 is the largest squarefree practical number with 3 prime factors and the largest prime factor is 13.

Frank M Jackson, Table of n, a(n) for n = 1..15
Wikipedia, Practical number and Squarefree integer

Cf. A002110, A005117, A005153, A265501.

lst={2}; Do[If[PrimeQ[f=DivisorSigma[1, Apply[Times, lst]]+1], AppendTo[lst, f], AppendTo[lst, NextPrime[f, -1]]], {8}]; lst
lst={2}; Do[AppendTo[lst, NextPrime[Times@@(#+1)&[lst]+2, -1]], {12}]; lst (* Frank M Jackson, May 29 2023 *)

A346245 Numbers k for which A003415(k) > k*A003557(k).

Original entry on oeis.org

30, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1218, 1230, 1290, 1302, 1410, 1554, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2370, 2490, 2670, 2730, 2910, 3030, 3090, 3210, 3270, 3390, 3570, 3810, 3930, 3990, 4110, 4170, 4290, 4470, 4530, 4710, 4830, 4890, 5010, 5190, 5370, 5430

Offset: 1

Antti Karttunen, Jul 19 2021

nonn

Numbers k such that A342001(k) > k.

Numbers k such that their arithmetic derivative (A003415(k)) is larger than A102631(k), k^2 / (squarefree kernel of k).

Positions of negative terms in A346244.
Cf. A003415, A003557, A102631, A342001.
Seems to have many common terms with A181629, A265501 and A286652.

cp's OEIS Frontend

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.

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A286652 Unitary practical numbers: numbers n such that every 1 <= k <= usigma(n) is a sum of distinct unitary divisors of n.

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A287173 Unitary practical numbers that are nonsquarefree.

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A268156 Smallest squarefree term of adjacent squarefree pairs in the sequence of practical numbers (A005153).

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A380328 2-dense squarefree numbers: Squarefree numbers whose divisors increase by factors of at most 2.

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A267422 Largest prime factor of the largest squarefree practical number comprising n prime factors.

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A346245 Numbers k for which A003415(k) > k*A003557(k).

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