what is the value "60" doing in A000028 ??
Meeussen Wouter (bkarnd)
wouter.meeussen at vandemoortele.com
Tue Sep 11 17:50:57 CEST 2001
same for "84" = 2 * 42 , and others.
Also remark that 76 does not appear in either sequence A000028 or A000379
and since it's 4*19 it belongs to A000379.
There is however a quite different construction that generates A000028 and
A000379
as they stand (with the lost "76" added). That one is based on the MoebiusMu
-analog for infinitary divisors, defined by forcing mu() into the role of
MoebiusMu :
the sum over the i-divisors of the function mu(n/i_d)*s[i_d] should
equal
the sum over the i-divisors of s[i_d]. Solve for mu().
Q: was the Name " a(n) is smallest number not of form a(i)a(j), i<j<n."
added at a later date?
*********** in plain 'Hungarian', this 'different construction' is
*******************
bitty[k_]:=Union[Flatten[Outer[Plus,Sequence@@
({0,#}&/@Union[(2^Range[0,Floor[Log[2,k]]])
Reverse[IntegerDigits[k,2]]])]]];
iDivisors[k_Integer]:= (Times @@ (First[it]^(#1 /. z -> List)) & ) /@
Flatten[Outer[z,Sequence @@ bitty/@Last[it=Transpose[FactorInteger[k]]],
1]] ;
iDivisors[1]:={1}
SumOveriDivisors[f_,n_Integer]:=Plus@@(Map[f[#1,n]&, iDivisors[n] ]);
inverseimoebius[seq_]:=SumOveriDivisors[ seq[[#1]] &, # ]& /@
Range[Length[seq]];
seq= s/@ Range[0, 632] ;
itry=SumOveriDivisors[\[Mu][#2/#1] seq[[#1]] &, # ]& /@ Range[Length[seq]]
;
inv=inverseimoebius[itry];
Array[\[Mu],632]/.Solve[Thread[inv== Array[s,633,0] ], Array[\[Mu],633]
]
the positions of "1" are A000028 and those of "-1" are A000379 (or the other
way 'round).
Wouter Meeussen
tel +32 (0)51 332 124
fax +32 (0)51 332 175
mail: wouter.meeussen at vandemoortele.com
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