A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.

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

Offset: 1

Author

Gus Wiseman, Oct 08 2024

Keywords

• nonn

Examples

			The a(3456) = 28 factorizations are:
  (4*8*9*12)  (4*9*96)    (36*96)   (3456)
              (8*9*48)    (4*864)
              (4*12*72)   (48*72)
              (4*16*54)   (54*64)
              (4*18*48)   (8*432)
              (4*24*36)   (9*384)
              (4*27*32)   (12*288)
              (4*8*108)   (16*216)
              (8*12*36)   (18*192)
              (8*16*27)   (24*144)
              (8*18*24)   (27*128)
              (9*12*32)   (32*108)
              (9*16*24)
              (12*16*18)
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of zeros are A005117 (squarefree numbers), complement A013929.

For squarefree instead of nonsquarefree we have A050326, non-strict A050320.

For prime-powers we have A050361, non-strict A000688.

For nonprime numbers we have A050372, non-strict A050370.

The version for partitions is A256012, non-strict A114374.

For perfect-powers we have A323090, non-strict A294068.

The non-strict version is A376657.

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, A303707, A322452, A375707, A376312.

Programs

JavaScript

function nextNonSquareFree(val){val+=1;for(let i=2;i*i<=val;i+=1){if(val%i==0&&val%(i*i)==0){return val}}return nextNonSquareFree(val)}function strictFactorCount(val,maxFactor){if(val==1){return 1}let sum=0;while(maxFactorDominic McCarty, Oct 19 2024

Mathematica

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],UnsameQ@@#&&NoneTrue[#,SquareFreeQ]&]],{n,100}] (* corrected by Gus Wiseman, Jun 27 2025 *)