%I #12 Mar 09 2022 16:32:53
%S 0,2,4,16,6,99,9,40,20,22,27,38,78,86,68,42,133,134,96,52,65,45,60,88,
%T 104,100,114,110,175,49,69,105,154,108,116,262,118,144,148,164,192,
%U 166,184,174,204,212,117,209,245,214,216,232,256,292,336,224,230,310
%N a(1) = 0; for n>0, a(n) is the smallest k in A350552 which is not yet a term and is such that k + a(n1) is in A350552.
%e a(2) = 2 = 10_2 and a(2)' = 2' = 1 = 1_2 have a single 1 in their binary expansion, a(1) + a(2) = 0 + 2 = 10_2 and (a(1) + a(2))' = (0 + 2)' = 2' = 1 = 1_2 have a single 1 in their binary expansion.
%e a(3) = 4 = 100_2, a(3)' = 4' = 4 = 100_2 have a single 1 in their binary expansion, and a(2) + a(3) = 2 + 4 = 6 = 110_2 and (a(2) + a(3))' = (2 + 4)' = 6' = 5 = 101_2, have two 1's in their binary expansion.
%t d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); v = Select[Range[0, 600], Equal @@ DigitCount[{#, d[#]}, 2, 1] &]; s = {}; found = True; i = {1}; While[i != {}, AppendTo[s, v[[i[[1]]]]]; i = FirstPosition[v, _?(! MemberQ[s, #] && MemberQ[v, s[[1]] + #] &)]]; s (* _Amiram Eldar_, Jan 27 2022 *)
%o (Magma) f:=func<n n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; ham:=func<a,bMultiplicity(Intseq(a,2),1) eq Multiplicity(Intseq(b,2),1)>; a:=[0]; for n in [2..57] do k:=1; while k in a or not ham(k,Floor(f(k))) or not ham(k+a[n1],Floor(f(k+a[n1]))) do k:=k+1; end while; Append(~a,k); end for; a;
%Y Cf. A000120, A003415, A350552.
%K nonn,base
%O 1,2
%A _Marius A. Burtea_, Jan 24 2022
