login
Number of strict integer factorizations of n into nonsquarefree factors > 1.
2

%I #10 Nov 04 2024 02:06:30

%S 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,

%T 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,

%U 0,0,0,3,0,0,1,1,0,0,0,2,1,0,0,1,0,0,0

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

%H Dominic McCarty, <a href="/A376679/b376679.txt">Table of n, a(n) for n = 1..10000</a>

%e The a(3456) = 28 factorizations are:

%e (4*8*9*12) (4*9*96) (36*96) (3456)

%e (8*9*48) (4*864)

%e (4*12*72) (48*72)

%e (4*16*54) (54*64)

%e (4*18*48) (8*432)

%e (4*24*36) (9*384)

%e (4*27*32) (12*288)

%e (4*8*108) (16*216)

%e (8*12*36) (18*192)

%e (8*16*27) (24*144)

%e (8*18*24) (27*128)

%e (9*12*32) (32*108)

%e (9*16*24)

%e (12*16*18)

%t Table[Length[Select[facs[n],UnsameQ@@#&&NoneTrue[#,SquareFreeQ]&]],{n,100}]

%o (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(maxFactor<val){maxFactor=nextNonSquareFree(maxFactor);if(val%maxFactor==0){sum+=strictFactorCount(val/maxFactor,maxFactor)}}return sum}let a="";for(let n=1;n<=100;n+=1){a+=strictFactorCount(n,0)+", "}console.log(a); // _Dominic McCarty_, Oct 19 2024

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

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

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

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

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

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

%Y The non-strict version is A376657.

%Y Nonsquarefree numbers:

%Y - A078147 (first differences)

%Y - A376593 (second differences)

%Y - A376594 (inflections and undulations)

%Y - A376595 (nonzero curvature)

%Y A000040 lists the prime numbers, differences A001223.

%Y A001055 counts integer factorizations, strict A045778.

%Y A005117 lists squarefree numbers, differences A076259.

%Y A317829 counts factorizations of superprimorials, strict A337069.

%Y Cf. A008480, A053797, A053806, A061398, A089259, A120992, A303707, A322452, A375707, A376312.

%K nonn

%O 1,32

%A _Gus Wiseman_, Oct 08 2024