A325782 -id:A325782 - cp's OEIS frontend

A325990 Numbers with more than one perfect factorization.

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A060476 in lacking 1 and having 432.

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Positions of terms > 1 in A325989.

Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

A327272 Smallest modulus of any (n+1) X (n+1) integer determinant whose top row is 1,2,2^2,...,2^n and whose rows are pairwise orthogonal.

Original entry on oeis.org

1, 5, 42, 425, 17050, 54600, 11468100

Offset: 1

Christopher J. Smyth, Sep 09 2019

An algorithm for generating a(n) is given in the Pinner and Smyth link, where more details about a(n) can be found.

Also, see file link below for {(n,a(n),matrix(n)),0 <= n <= 6}, where matrix(n) has minimal modulus determinant equal to a(n) among (n+1) X (n+1) matrices with top row 1,2,2^2,...,2^n and all rows orthogonal.

			a(2) =42 since det([[1,2,4],[2,-3,1],[2,1,-1]]) = 42 and is the smallest positive determinant with top row [1,2,2^2] and all entries integers, and rows orthogonal.

Chris Pinner and Chris Smyth, Lattices of minimal index in Z^n having an orthogonal basis containing a given basis vector
Christopher J. Smyth, List of n, a(n) and associated matrix for 0 <= n <= 6

Subsequence of A327267-- see comments; A327269 is similar, but determinant's top row is 1,2,...,n; A327271 is similar, but determinant's top row consists of n 1's.

a(n) = A327267(Product_{k=0..n} prime(2^k)) = A327267(A325782(n+1)).

A327403 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum stable divisor (A327393, A327402).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476. The maximum stable divisor of n is A327393(n).

			We have 798 -> 42 -> 6 -> 2 -> 1, so a(798) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Index entries for sequences related to prime indices in the factorization of n.

See link for additional cross-references.
Positions of first appearance of each integer are A325782.
Cf. A000005, A303362, A323014, A327393, A327402.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[FixedPointList[#/Max[Select[Divisors[#],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]]&,n]]-2,{n,100}]

A327403(n) = for(k=0,oo,my(nextn=A327402(n)); if(nextn==n,return(k)); n = nextn); \\ Antti Karttunen, Jan 28 2025

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

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A325990 Numbers with more than one perfect factorization.

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A327272 Smallest modulus of any (n+1) X (n+1) integer determinant whose top row is 1,2,2^2,...,2^n and whose rows are pairwise orthogonal.

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A327403 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum stable divisor (A327393, A327402).

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