cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

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Author

Frank M Jackson, Jan 10 2016

Keywords

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

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

A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 20, 16, 18, 40, 28, 32, 24, 80, 56, 30, 64, 36, 100, 112, 60, 42, 128, 48, 160, 196, 90, 84, 66, 256, 54, 200, 224, 120, 126, 132, 78, 512, 72, 320, 392, 150, 168, 198, 156, 88, 1024, 96, 400, 448, 180, 252, 264, 234, 176, 104, 2048, 108, 500, 784, 240, 294, 396, 312, 352, 208, 140, 4096, 144, 640, 896, 270, 336, 528, 468, 704, 416, 280, 204, 8192
Offset: 1

Views

Author

Frank M Jackson, Dec 30 2024

Keywords

Comments

A permutation of the practical numbers.
This sequence is presented as an array of rows. The first row contains a single term of value 1. Subsequent rows are infinite sequences and are presented as a square array by listing the antidiagonals downwards that is 1: 2; 4,6; 8,12,20; etc.
The first column contains the primitive practical numbers A267124; each row lists all practical numbers (A005153) having the same primitive practicle progenitor and which is the first term in each row. See A379325 comments for further details. If T[1,m] is squarefree then the row is identical to the same squarefree row in A284457.
Every primitive practical number A267124(n) is the progenitor of a disjoint subsequence of the practical numbers. If the PP column represents the sequence of primitive practical numbers A267124, the table below give the 7 initial terms of the disjoint sequences of practical numbers A005153 generated by the initial 7 terms of the sequence of primitive practical numbers.
PP: Disjoint subsequence of A005153
-- -------------------------------
1: 1
2: 2, 4, 8, 16, 32, 64,128, . . .- A000079 with offset 1,1
6: 6, 12, 18, 24, 36, 48, 54, . . .- A033845
20: 20, 40, 80,100,160,200,320, . . .
28: 28, 56,112,196,224,392,448, . . .
30: 30, 60, 90,120,150,180,240, . . .- A143207
42: 42, 84,126,168,252,294,336
...
Row 1 is T[1,1] = 1 and only has one term in the subsequence.
Row 7 is T[1,7] = 2*3*7; T[2,7] = 2^2*3*7; T[3,7] = 2*3^2*7; T[4,7] = 2^3*3*7; T[5,7] = 2^2*3^2*7, etc.

Examples

			a(14) = 80 and it is T[3,4] = 2^4*5. Its primitive progenitor is 20 = 2^2*5 and its equivalence class are the terms of row 4.

Links

Crossrefs

Cf. A000079, A005117, A005153, A007947, A033845, A143207, A267124, A284457, A379325.

Programs

  • Mathematica
    (* See link above *)

A379698 Exponent of highest power of 2 that divides the n-th practical number A005153(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 2, 1, 5, 2, 3, 1, 4, 1, 3, 2, 6, 1, 3, 1, 4, 2, 3, 1, 5, 2, 3, 2, 4, 3, 1, 7, 2, 2, 4, 1, 2, 5, 1, 3, 4, 2, 6, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 1, 4, 2, 8, 2, 3, 1, 4, 2, 3, 5, 1, 2, 4, 1, 2, 3, 6, 2, 1, 4, 2, 1, 2, 5, 3, 2, 4, 1, 2, 7, 1, 3, 2, 4, 3, 1, 5, 2, 4, 3, 6, 1, 3, 2, 1, 4, 2, 2, 5, 1, 4, 2, 3, 1, 9, 3, 1, 4, 2, 2, 5, 1
Offset: 1

Views

Author

Frank M Jackson, Dec 30 2024

Keywords

Comments

The relationship between a practical number and highest power of 2 that divides it provides an equivalence relation. Practical numbers whose prime factorization have the same exponent of 2.

Examples

			a(9) = 2. A005153(9) = 20 and the greatest power of 2 that divides 20 is 2.

Links

Crossrefs

Cf. A005153, A007814, A379325.

Programs

  • Mathematica
    plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; lst={}; Do[If[OddQ[plst[[n]]], AppendTo[lst, 0], AppendTo[lst, Last@First@FactorInteger[plst[[n]]]]], {n, 1, 100}]; lst

Formula

a(n) = A007814(A005153(n)).
Showing 1-3 of 3 results.

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