A294068 -id:A294068 - cp's OEIS frontend

A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 11 2018

nonn

All terms are squarefree numbers. First differs from A304328 at a(500) = 1, A304328(500) = 4.

			f maps 500 -> 4 -> 1 -> 1, so a(500) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001597, A001694, A005117, A007916, A052485, A052486, A059404, A091050, A160400, A203025, A294068, A303707, A304327, A304328.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
op[n_]:=n/Last[Select[Divisors[n],!radQ[#]&]];
Table[FixedPoint[op,n],{n,200}]

a(n)={while(1, my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); if(m==1, return(n)); n/=m)} \\ Andrew Howroyd, Aug 26 2018

A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

Original entry on oeis.org

1, 1, 1, 3, 14, 48, 384, 1308, 40288, 933848, 21077680, 75690016, 5471262080, 7964665440, 54595767744, 17948164982144, 3454946386353664, 5010658671663616, 723456523262697984, 950502767770273280, 165679731871366906880, 8443707247468681128448

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(1) = 1 through a(4) = 14 chains (with n! prepended):
  1  2/1  6/1    24/1
          6/2/1  24/2/1
          6/3/1  24/3/1
                 24/4/1
                 24/8/1
                 24/12/1
                 24/4/2/1
                 24/8/2/1
                 24/8/4/1
                 24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/8/4/2/1
                 24/12/4/2/1

A336571 is the generalization to not just factorial numbers.
A337104 is the version for chains containing n!.
A000005 counts divisors.
A001055 counts factorizations.
A032741 counts proper divisors.
A071625 counts distinct prime multiplicities.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336424 counts factorizations using A130091.
A336425 counts divisible pairs of divisors of n!, both in A130091.
Cf. A002033, A098859, A124010, A294068, A327499, A327500, A327523, A337071.

chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[n!],{n,0,5}]

a(n) = A337104(n) whenever A337104(n) != 0.

a(n) = A336571(n!).

A376657 Number of integer factorizations of n into nonsquarefree factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 07 2024

nonn

			The a(n) factorizations for n = 16, 64, 72, 144, 192, 256, 288:
  (16)   (64)     (72)    (144)    (192)     (256)      (288)
  (4*4)  (8*8)    (8*9)   (4*36)   (4*48)    (4*64)     (4*72)
         (4*16)   (4*18)  (8*18)   (8*24)    (8*32)     (8*36)
         (4*4*4)          (9*16)   (12*16)   (16*16)    (9*32)
                          (12*12)  (4*4*12)  (4*8*8)    (12*24)
                          (4*4*9)            (4*4*16)   (16*18)
                                             (4*4*4*4)  (4*8*9)
                                                        (4*4*18)

For prime-powers we have A000688.
Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050320, strict A050326.
For nonprime numbers we have A050370.
The version for partitions is A114374.
For perfect-powers we have A294068.
For non-perfect-powers we have A303707.
For non-prime-powers we have A322452.
The strict case is A376679.
Nonsquarefree numbers:
- A078147 (first differences)
- A376593 (second differences)
- A376594 (inflections and undulations)
- A376595 (nonzero curvature)
A000040 lists the prime numbers, differences A001223.
A001055 counts integer factorizations, strict A045778.
A005117 lists squarefree numbers, differences A076259.
A317829 counts factorizations of superprimorials, strict A337069.
Cf. A008480, A053797, A053806, A061398, A089259, A120992, A124010, A182853, A373198, A375707, A376312.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],NoneTrue[SquareFreeQ]]],{n,100}]

A339553 Number of ordered factorizations of n into perfect powers > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 8

Offset: 1

Ilya Gutkovskiy, Dec 08 2020

nonn

Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A050363, A052485 (positions of 0's), A075802, A294068.

a[n_] := If[n == 1, n, Sum[If[d < n, Boole[GCD @@ FactorInteger[n/d][[All, 2]] > 1] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 128}]

a(1) = 1; a(n) = Sum_{d|n, dA075802(n/d) * a(d).

cp's OEIS Frontend

A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

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A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

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A376657 Number of integer factorizations of n into nonsquarefree factors > 1.

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A339553 Number of ordered factorizations of n into perfect powers > 1.

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Mathematica

Formula