A350553 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).

0, 2, 6, 22, 27, 110, 175, 502, 894, 2037, 3775, 8182, 24558, 49142, 98286, 196598, 655323, 524278, 2088950, 2097142, 6291438, 16515062, 15728575, 62914175, 100663278, 134217718, 528482294, 939524086, 2145386486, 3221225454, 11811159998, 8589934582, 47244640246

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..32.

Example

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

Mathematica

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 *)

Prog

(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;
(PARI) d(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = my(k=2^n-1); while ((hammingweight(k) != n) || (hammingweight(d(k)) != n), k++); k; \\ Michel Marcus, Jan 25 2022

Crossrefs

Cf. A000120, A003415, A350552.

Sequence in context: A085286 A062085 A284632 * A325020 A335131 A147800

Adjacent sequences: A350550 A350551 A350552 * A350554 A350555 A350556

Keyword

nonn,base

Author

Marius A. Burtea, Jan 24 2022

Extensions

a(30)-a(32) from Jinyuan Wang, Jan 27 2022

Status

approved