A350553 - OEIS

%I #23 Jan 28 2022 20:28:08

%S 0,2,6,22,27,110,175,502,894,2037,3775,8182,24558,49142,98286,196598,

%T 655323,524278,2088950,2097142,6291438,16515062,15728575,62914175,

%U 100663278,134217718,528482294,939524086,2145386486,3221225454,11811159998,8589934582,47244640246

%N a(n) is the smallest number k for which A000120(k) = A000120(k') = n, or -1 if no such k exists, where k' is the arithmetic derivative of k (A003415).

%C Conjecture. a(n) >= 0 for all n.

%e 0' = 0, A000120(0) = A000120(0') = 0, so a(0) = 0.

%e 1' = 0, A000120(1) = 1, A000120(1') = 0. 2 = 10_2, 2' = 1 = 1_0, so A000120(2) = A000120(2') = 1 and a(1) = 2.

%t bw[n_] := DigitCount[n, 2, 1]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); seq[m_, nmax_] := Module[{s = Table[-1, {m + 1}], c = 0, n = 0, i}, While[c < m + 1 && n < nmax, i = bw[n] + 1; If[i <= m && s[[i]] < 0, If[bw[d[n]] + 1 == i, c++; s[[i]] = n]]; n++]; TakeWhile[s, # > -1 &]]; seq[16, 10^6] (* _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)]])>; a:=[]; for n in [0..22] do k:=2^n-1 ; while &+Intseq(k,2) ne n or  &+Intseq( Floor(f(k)),2)  ne n do k:=k+1; end while; Append(~a,k*1);  end for; a;

%o (PARI) d(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415

%o a(n) = my(k=2^n-1); while ((hammingweight(k) != n) || (hammingweight(d(k)) != n), k++); k; \\ _Michel Marcus_, Jan 25 2022

%Y Cf. A000120, A003415, A350552.

%K nonn,base

%O 0,2

%A _Marius A. Burtea_, Jan 24 2022

%E a(30)-a(32) from _Jinyuan Wang_, Jan 27 2022