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Number of augmented amicable pairs (m,n) with m<n and for which m<=10^k, k=1,2,3,...
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%I #4 Mar 30 2012 18:41:04

%S 0,0,0,1,4,9,36,84,188,420,930,1931

%N Number of augmented amicable pairs (m,n) with m<n and for which m<=10^k, k=1,2,3,...

%H J. M. Pedersen, <a href="http://amicable.homepage.dk/knwnc2.htm">Known amicable pairs</a>.

%F An augmented amicable pair (m,n) is a pair of integers m, n with m<n and for which sigma(m)=sigma(n)=m+n-1

%e a(6)=9 because there are 9 augmented amicable pairs with m<=10^6

%t s[n_]:=DivisorSigma[1,n]-n; AugmentedAmicableNumberQ[n_]:=If[s[s[n]+1]==n-1 && !DivisorSigma[1,n]==2n-1,True,False]; AugmentedAmicablePairList[ k_]:=(bnlist=Select[Range[k], AugmentedAmicableNumberQ[ # ]&]; newprlist= Table[Sort[{bnlist[[n]],s[bnlist[[n]]]+1}],{n,1,Length[bnlist]}]; augamprlist=Union[newprlist,newprlist]); data=AugmentedAmicablePairList[10^7]; Table[Length[Select[data,First[ # ]<10^k &]],{k,1,7}]

%Y Cf. A007992, A015630.

%K hard,nonn

%O 1,5

%A _Ant King_, Dec 20 2006