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%I #35 Apr 19 2021 17:11:09
%S 0,1,1,3,1,3,1,6,3,3,1,8,1,3,3,12,1,8,1,8,3,3,1,17,3,3,6,8,1,10,1,20,
%T 3,3,3,22,1,3,3,17,1,10,1,8,8,3,1,34,3,8,3,8,1,17,3,17,3,3,1,27,1,3,8,
%U 35,3,10,1,8,3,10,1,46,1,3,8,8,3,10,1,34,12,3,1,27,3,3,3,17,1,27,3,8,3,3,3
%N Total number of elements in all factorizations of n with all factors > 1.
%C From _Gus Wiseman_, Apr 18 2021: (Start)
%C Number of ways to choose a factor index or position in a factorization of n. The version selecting a factor value is A339564. For example, the factorizations of n = 2, 4, 8, 12, 16, 24, 30 with a selected position (in parentheses) are:
%C ((2)) ((4)) ((8)) ((12)) ((16)) ((24)) ((30))
%C ((2)*2) ((2)*4) ((2)*6) ((2)*8) ((3)*8) ((5)*6)
%C (2*(2)) (2*(4)) (2*(6)) (2*(8)) (3*(8)) (5*(6))
%C ((2)*2*2) ((3)*4) ((4)*4) ((4)*6) ((2)*15)
%C (2*(2)*2) (3*(4)) (4*(4)) (4*(6)) (2*(15))
%C (2*2*(2)) ((2)*2*3) ((2)*2*4) ((2)*12) ((3)*10)
%C (2*(2)*3) (2*(2)*4) (2*(12)) (3*(10))
%C (2*2*(3)) (2*2*(4)) ((2)*2*6) ((2)*3*5)
%C ((2)*2*2*2) (2*(2)*6) (2*(3)*5)
%C (2*(2)*2*2) (2*2*(6)) (2*3*(5))
%C (2*2*(2)*2) ((2)*3*4)
%C (2*2*2*(2)) (2*(3)*4)
%C (2*3*(4))
%C ((2)*2*2*3)
%C (2*(2)*2*3)
%C (2*2*(2)*3)
%C (2*2*2*(3))
%C (End)
%D Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
%D Amarnath Murthy, Length and extent of Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
%H Alois P. Heinz, <a href="/A066637/b066637.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from T. D. Noe)
%e a(12) = 8: there are 4 factorizations of 12: (12), (6*2), (4*3), (3*2*2) having 1, 2, 2, 3 elements respectively, a total of 8.
%p # Return a list of lists which are factorizations (product representations)
%p # of n. Within each sublist, the factors are sorted. A minimum factor in
%p # each element of sublists returned can be specified with 'mincomp'.
%p # If mincomp=2, the number of sublists contained in the list returned is A001055(n).
%p # Example:
%p # n=8 and mincomp=2 return [[2,2,2],[4,8],[8]]
%p listProdRep := proc(n,mincomp)
%p local dvs,resul,f,i,j,rli,tmp ;
%p resul := [] ;
%p # list returned is empty if n < mincomp
%p if n >= mincomp then
%p if n = 1 then
%p RETURN([1]) ;
%p else
%p # compute the divisors, and take each divisor
%p # as a head element (minimum element) of one of the
%p # sublists. Example: for n=8 use {1,2,4,8}, and consider
%p # (for mincomp=2) sublists [2,...], [4,...] and [8].
%p dvs := numtheory[divisors](n) ;
%p for i from 1 to nops(dvs) do
%p # select the head element 'f' from the divisors
%p f := op(i,dvs) ;
%p # if this is already the maximum divisor n
%p # itself, this head element is the last in
%p # the sublist
%p if f =n and f >= mincomp then
%p resul := [op(resul),[f]] ;
%p elif f >= mincomp then
%p # if this is not the maximum element
%p # n itself, produce all factorizations
%p # of the remaining factor recursively.
%p rli := procname(n/f,f) ;
%p # Prepend all the results produced
%p # from the recursion with the head
%p # element for the result.
%p for j from 1 to nops(rli) do
%p tmp := [f,op(op(j,rli))] ;
%p resul := [op(resul),tmp] ;
%p od ;
%p fi ;
%p od ;
%p fi ;
%p fi ;
%p resul ;
%p end:
%p A066637 := proc(n)
%p local f,d;
%p a := 0 ;
%p for d in listProdRep(n,2) do
%p a := a+nops(d) ;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Jul 11 2013
%p # second Maple program:
%p with(numtheory):
%p b:= proc(n, k) option remember; `if`(n>k, 0, [1$2])+
%p `if`(isprime(n), 0, (p-> p+[0, p[1]])(add(
%p `if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n})))
%p end:
%p a:= n-> `if`(n<2, 0, b(n$2)[2]):
%p seq(a(n), n=1..120); # _Alois P. Heinz_, Feb 12 2019
%t g[1, r_] := g[1, r]={1, 0}; g[n_, r_] := g[n, r]=Module[{ds, i, val}, ds=Select[Divisors[n], 1<#<=r&]; val={0, 0}+Sum[g[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]; val+{0, val[[1]]}]; a[n_] := g[n, n][[2]]; a/@Range[95] (* g[n, r] = {c, f}, where c is the number of factorizations of n with factors <= r and f is the total number of factors in them. - _Dean Hickerson_, Oct 28 2002 *)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];Table[Sum[Length[fac],{fac,facs[n]}],{n,50}] (* _Gus Wiseman_, Apr 18 2021 *)
%Y The version for normal multisets is A001787.
%Y The version for compositions is A001792.
%Y The version for partitions is A006128 (strict: A015723).
%Y Choosing a value instead of position gives A339564.
%Y A000070 counts partitions with a selected part.
%Y A001055 counts factorizations.
%Y A002033 and A074206 count ordered factorizations.
%Y A067824 counts strict chains of divisors starting with n.
%Y A336875 counts compositions with a selected part.
%Y Cf. A000005, A000041, A045778, A050336, A066186, A162247, A264401, A281116, A292504, A292886, A322794.
%K nonn
%O 1,4
%A _Amarnath Murthy_, Dec 28 2001
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