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A212664 Least k with precisely n partitions k = x + y satisfying x > 0 and k’ = x’ + y’, where k’, x’, y’ are the arithmetic derivatives of k, x, y.

%I A212664 #18 May 25 2012 14:40:30
%S A212664 3,39,213,903,2379,2343,6545,12325,15015,16107,45045,134225,80535,
%T A212664 142545,205205,255255,346035,533715,615615,645645,997815,1601145,
%U A212664 1369095,1936935
%N A212664 Least k with precisely n partitions k = x + y satisfying x > 0 and k’ = x’ + y’, where k’, x’, y’ are the arithmetic derivatives of k, x, y.
%e A212664 n=2343, x=162, y=2181 and 2343=162+2181; n’=1027, x’=297, y’=730 and 1027=297+730.
%e A212664 n=2343, x=308, y=2035 and 2343=308+2035; n’=1027, x’=380, y’=647 and 1027=+380+647.
%e A212664 n=2343, x=377, y=1966 and 2343=377+1966; n’=1027, x’=42, y’=985 and 1027=42+985.
%e A212664 n=2343, x=484, y=1859 and 2343=484+1859; n’=1027, x’=572, y’=455 and 1027=572+455.
%e A212664 n=2343, x=505, y=1838 and 2343=505+1838; n’=1027, x’=106, y’=921 and 1027=106+921.
%e A212664 n=2343, x=781, y=1562 and 2343=781+1562; n’=1027, x’=82, y’=945 and 1027=82+945.
%p A212664 with(numtheory);
%p A212664 A212664:=proc(q)
%p A212664 local a, b, c, d, f, i, j, n, p, pfs, v;
%p A212664 v:=array(1..100); for n from 1 to 100 do v[n]:=0; od;
%p A212664 a:=0;
%p A212664 for n from 1 to q do
%p A212664   pfs:=ifactors(n)[2]; c:=n*add(op(2,p)/op(1,p),p=pfs); b:=0;
%p A212664   for i from 1 to trunc(n/2) do
%p A212664     pfs:=ifactors(i)[2]; d:=i*add(op(2,p)/op(1,p),p=pfs);
%p A212664     pfs:=ifactors(n-i)[2]; f:=(n-i)*add(op(2,p)/op(1,p),p=pfs);
%p A212664     if c=d+f then b:=b+1; fi; od;
%p A212664     if b=a+1 then a:=b; print(b,n); j:=1;
%p A212664        while v[b+j]>0 do a:=b+j; print(b,v[b+j]); j:=j+1; od;
%p A212664     else if b>a+1 then if v[b]=0 then v[b]:=n; fi; fi; fi;
%p A212664 od; end:
%p A212664 A212664(100000);
%Y A212664 Cf. A003415, A211223-A211225, A212662, A212663.
%K A212664 nonn,more,hard
%O A212664 1,1
%A A212664 _Paolo P. Lava_, May 23 2012
%E A212664 a(12)-a(24) from _Donovan Johnson_, May 25 2012