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A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.

%I A363464 #21 Aug 05 2023 22:04:21
%S A363464 6,9,10,14,18,20,21,22,24,25,33,34,35,38,40,42,44,48,49,52,62,65,66,
%T A363464 68,69,70,76,80,84,88,91,93,94,96,100,104,110,115,117,118,121,132,133,
%U A363464 134,138,140,143,144,145,148,152,155,158,164,174,182,185,186,188,192
%N A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.
%C A363464 If p > 2 is in A092506 then m = 2*p and u = 4*p are terms. Indeed, if p = 2^k + 1, k >= 1, m = 2*(2^k + 1) = 2^(k+1) + 2^1 has two 1's in its binary expansion, and m' = p+2 = 2^k + 3 = 2^k + 2^1 + 1 has three 1's in its binary expansion. Similarly u = 4*(2^k + 1) = 2^(k+2) + 2^2 and u' = 4*p + 4 = 2^(k+2) + 2^3.
%C A363464 If p is in A057733 then the number m = 2*p is a term. Indeed, if p = 2^k + 3, k >= 1, m = 2*(2^k + 3) = 2^(k+1) + 2^2 + 2 has three 1's in its binary expansion, and m' = p+2 = 2^k + 5 = 2^k + 2^2 + 1 has three 1's in its binary expansion.
%C A363464 If p > 7 is in A057733 then the number m = 4*p is a term. Indeed, if p = 2^k + 3, k >= 3, m = 4*(2^k + 3) = 2^(k+2) + 2^3 + 2 has three 1's in its binary expansion, and m' = 4*(p + 1) = 4*(2^k + 4) = 2^(k+2) + 2^4 has two 1's in its binary expansion.
%C A363464 If p is in A123250 then the number m = 4*p is a term. Indeed, if p = 2^k + 5, k >= 1, m = 4*(2^k + 5) = 2^(k+2) + 2^4 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 6) = 2^(k+2) + 2^4 + 2^2 has three 1's in its binary expansion.
%C A363464 If p is in A104070 then the number m = 4*p is a term. Indeed, if p = 2^k + 9, k >= 1, m = 4*(2^k + 9) = 2^(k+2) + 2^5 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 10) = 2^(k+2) + 2^5 + 2^3 has three 1's in its binary expansion.
%e A363464 6 = 110_2 has two 1's, 6' = 5 = 101_2 has two 1's, so 6 is a term.
%e A363464 9 = 101_2 has two 1's, 9' = 6 = 110_2 has two 1's, so 9 is a term.
%e A363464 10 = 1010_2 has two 1's, 10' = 7 = 111_2 has three 1's, so 10 is a term.
%e A363464 18 = 10010_2 has two 1's, 18' = 21 = 10101_2 has three 1's, so 18 is a term.
%t A363464 pernQ[n_] := PrimeQ[DigitCount[n, 2, 1]]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200], And @@ pernQ[{#, d[#]}] &] (* _Amiram Eldar_, Jul 10 2023 *)
%o A363464 (Magma) fp:=func<n|IsPrime(Multiplicity(Intseq(n,2),1)) >; f:=func<n |n le 1
%o A363464 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in
%o A363464 [1..#Factorisation(n)]])>; [n:n in [1..200]| fp(n) and fp(Floor(f(n)))];
%Y A363464 Cf. A000120, A003415, A052294, A057733, A081092, A092506, A104070, A123250.
%K A363464 nonn,base
%O A363464 1,1
%A A363464 _Marius A. Burtea_, Jul 08 2023