A357840 Numbers k in A018900 with arithmetic derivative k' (A003415) in A018900.

6, 9, 20, 40, 65, 68, 96, 144, 192, 528, 576, 1028, 4097, 73728, 81920, 262148, 557056, 6291456, 9437184, 12582912, 201326592, 335544320, 2415919104, 1374389534720, 11258999068426240, 90071992547409920, 648518346341351424, 78398662313265594368, 116056878683004400771792896

Offset: 1

Author

Marius A. Burtea, Oct 20 2022

Keywords

• nonn
• base

Comments

Numbers with Hamming weight 2 that have the arithmetic derivative with Hamming weight 2.

If p is in A019434 then m = 4*p is a term. Indeed p = 2^k + 1 and m' = 4*p + 4 = 4 (p + 1) = 4*(2^k + 2) = 8*(2^(k - 1) +1) = 2^(k + 2) + 2^3 and m, m' have only two of 1 in the expansion binary.

Numbers of the form m = 5*2^s, where s = (16^k - 1)/5, s >= 1, are terms. Indeed m = 4*2^s + 2^s = 2^(s + 2) + 2^s and m' = 2^s + 5*s*2^(s - 1) = 2^(s - 1)*(5*s + 2) = 2^(s - 1)*(16^k + 1) = 2^(4*k + s - 1) + 2^(s - 1), so m is a term.

Numbers of the form m = 3*2^s, where s = (4^k - 1)/3, s >= 1, are terms. Indeed m = 2*2^s + 2^s = 2^(s + 1) + 2^s and m' = 2^s + 3*s*2^(s - 1) = 2^(s - 1)*(3*s + 2) = 2^(s - 1)*(4^k + 1) = 2^(2*k + s - 1) + 2^(s - 1), so m is a term.

Examples

			6 = 110_2 = 2^2 + 2^1 and 6' = 5 = 101_2 = 2^2 + 2^0, so 6 is a term.
9 = 1001_2 = 2^3 + 2^0 and 9' = 6 = 110_2, so 9 is a term.
20 = 10100_2 = 2^4 + 2^2 and 20' = 24 = 11000_2 = 2^4 + 2^3, so 20 is a term.
		

Crossrefs

Cf. A003415, A018900, A019434.

Programs

Magma

f1:=func; f:=func; a:=[]; for n in [1..75] do sn:=[2^n+2^k: k in [0..n-1]]; for i in [1..#sn] do if f1(sn[i]) and f1(Floor(f(sn[i]))) then Append(~a,sn[i]); end if; end for; end for; a;

Mathematica

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Union[Plus @@@ (2^Subsets[Range[0, 86], {2}])], Count[IntegerDigits[d[#], 2], 1] == 2 &] (* Amiram Eldar, Oct 21 2022 *)

PARI

ish2(n) = hammingweight(n)==2; \\ A018900
ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = ish2(m) && ish2(ad(m)); \\ Michel Marcus, Oct 23 2022