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

%I A376679 #17 Jun 27 2025 08:38:43
%S A376679 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 A376679 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 A376679 0,0,0,3,0,0,1,1,0,0,0,2,1,0,0,1,0,0,0
%N A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.
%H A376679 Dominic McCarty, <a href="/A376679/b376679.txt">Table of n, a(n) for n = 1..10000</a>
%e A376679 The a(3456) = 28 factorizations are:
%e A376679   (4*8*9*12)  (4*9*96)    (36*96)   (3456)
%e A376679               (8*9*48)    (4*864)
%e A376679               (4*12*72)   (48*72)
%e A376679               (4*16*54)   (54*64)
%e A376679               (4*18*48)   (8*432)
%e A376679               (4*24*36)   (9*384)
%e A376679               (4*27*32)   (12*288)
%e A376679               (4*8*108)   (16*216)
%e A376679               (8*12*36)   (18*192)
%e A376679               (8*16*27)   (24*144)
%e A376679               (8*18*24)   (27*128)
%e A376679               (9*12*32)   (32*108)
%e A376679               (9*16*24)
%e A376679               (12*16*18)
%t A376679 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A376679 Table[Length[Select[facs[n],UnsameQ@@#&&NoneTrue[#,SquareFreeQ]&]],{n,100}] (* corrected by _Gus Wiseman_, Jun 27 2025 *)
%o A376679 (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 A376679 Positions of zeros are A005117 (squarefree numbers), complement A013929.
%Y A376679 For squarefree instead of nonsquarefree we have A050326, non-strict A050320.
%Y A376679 For prime-powers we have A050361, non-strict A000688.
%Y A376679 For nonprime numbers we have A050372, non-strict A050370.
%Y A376679 The version for partitions is A256012, non-strict A114374.
%Y A376679 For perfect-powers we have A323090, non-strict A294068.
%Y A376679 The non-strict version is A376657.
%Y A376679 Nonsquarefree numbers:
%Y A376679 - A078147 (first differences)
%Y A376679 - A376593 (second differences)
%Y A376679 - A376594 (inflections and undulations)
%Y A376679 - A376595 (nonzero curvature)
%Y A376679 A000040 lists the prime numbers, differences A001223.
%Y A376679 A001055 counts integer factorizations, strict A045778.
%Y A376679 A005117 lists squarefree numbers, differences A076259.
%Y A376679 A317829 counts factorizations of superprimorials, strict A337069.
%Y A376679 Cf. A008480, A053797, A053806, A061398, A089259, A120992, A303707, A322452, A375707, A376312.
%K A376679 nonn
%O A376679 1,32
%A A376679 _Gus Wiseman_, Oct 08 2024