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A293252 Numbers k such that k = x + y, k' = x' + y' and k'' = x'' + y'', where k' and k'' are the first and second arithmetic derivatives of k.

%I A293252 #14 Oct 09 2017 06:15:09
%S A293252 3,778,1331,1575,1589,3111,5368,14060,17649,17714,23232,33813,34353,
%T A293252 36234,52936,53391,66375,74544,80938,88945,93475,94905,97470,98434,
%U A293252 156816,180804,207754,229502,238830,267120,274065,357318,367921,400500,406700,411872,418037
%N A293252 Numbers k such that k = x + y, k' = x' + y' and k'' = x'' + y'', where k' and k'' are the first and second arithmetic derivatives of k.
%C A293252 A226779(n) + 1 are terms of the sequence: for these numbers the relation stands for any following derivative because n = 1 + (n-1), n' = 0 + (n-1)' and n' = (n-1)' by definition. Apart 3, no other prime p can be in the sequence because p = x + y implies p' = 1 = x' + y' that is impossible (for 3 we have 3 = 1 + 2 and 3' = 1 = 1' + 2' = 0 + 1). Similarly, x and y cannot be both primes.
%C A293252 Is there any number that admits two or more different partitions?
%H A293252 Paolo P. Lava, <a href="/A293252/a293252.txt">List of k, x, and y</a>
%e A293252 1331 = 198 + 1133, 1331' = 363 = 198' + 1133' = 249 + 114, 1331'' = 187 = 198'' + 1133'' = 86 + 101.
%p A293252 with(numtheory): P:=proc(q) local a,b,c,k,n,p; for n from 1 to q do
%p A293252 for k from 1 to trunc(n/2) do a:=k*add(op(2,p)/op(1,p),p=ifactors(k)[2]);
%p A293252 b:=(n-k)*add(op(2,p)/op(1,p),p=ifactors(n-k)[2]); c:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); if c=a+b then a:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); b:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]); c:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]);
%p A293252 if c=a+b then print(n); break; fi; fi; od; od; end: P(10^5);
%t A293252 f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range[2000], Function[k, Count[IntegerPartitions[k, {2}], _?(And[f@ k == f@ #1 + f@ #2, Nest[f, k, 2] == Nest[f, #1, 2] + Nest[f, #2, 2]] & @@ # &)] > 0]] (* _Michael De Vlieger_, Oct 08 2017 *)
%Y A293252 Cf. A003415, A226779.
%K A293252 nonn
%O A293252 1,1
%A A293252 _Paolo P. Lava_, Oct 04 2017
%E A293252 a(25)-a(37) from _Giovanni Resta_, Oct 05 2017