A377377 a(n) is the quotient of the practical number A005153(n) divided by its largest divisor that is primitive practical.

1, 1, 2, 1, 4, 2, 8, 3, 1, 4, 1, 1, 16, 6, 2, 1, 8, 9, 2, 2, 32, 1, 12, 1, 4, 2, 1, 3, 16, 5, 1, 18, 4, 4, 3, 64, 2, 1, 24, 5, 2, 8, 27, 4, 2, 6, 32, 7, 3, 10, 1, 2, 1, 36, 1, 8, 1, 3, 8, 6, 128, 1, 3, 9, 1, 1, 2, 48, 7, 10, 1, 1, 1, 3, 16, 54, 1, 8, 1, 1, 1, 4, 12, 1, 1, 9, 1, 64, 1, 14, 6, 20, 2, 1, 4, 2, 72, 2, 16, 15, 2, 1, 1, 1, 6, 1, 16, 81, 1, 25, 12

Offset: 1

Author

Frank M Jackson, Oct 26 2024

Keywords

• nonn

Comments

Every practical number > 1 contains at least one primitive practical divisor because they are all even and 2 is primitive practical. Also 1 is primitive practical. If the largest primitive practical factor of the practical number A005153(n) is p then a(n)*p = A005153(n). Whenever a(n) = 1, A005153(n) is also primitive practical.

Examples

			a(13) = 16 because the practical number A005153(13) = 32 = 2*16 and 2 is its largest primitive practical factor.
a(14) = 6 because the practical number A005153(14) = 36 = 6*6 and 6 is its largest primitive practical factor.
		

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005153, A267124.

Programs

Mathematica

lst1=Last/@ReadList["https://oeis.org/A267124/b267124.txt", {Number, Number}]; lst2=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; getm[p_] := Module[{plst=Select[lst1, #<=p &], k, l, n=0}, l=Length@plst; If[Last@plst==p, Return[1]]; While[!IntegerQ[k=p/plst[[l-n]]], n++]; k]; Table[getm[lst2[[n]]], {n, 1, 100}]