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A350552 Numbers k for which A000120(k) = A000120(k'), where k' is arithmetic derivative of k (A003415). 2
0, 2, 4, 6, 9, 16, 20, 22, 27, 38, 40, 42, 45, 49, 52, 60, 65, 68, 69, 78, 86, 88, 96, 99, 100, 104, 105, 108, 110, 114, 116, 117, 118, 133, 134, 144, 148, 154, 164, 166, 174, 175, 184, 192, 204, 209, 212, 214, 216, 224, 225, 230, 232, 244, 245, 256, 259, 262 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The sequence is infinite because (p^p)' = p^p, for any p prime number (A051674).
If m = 2^k (A000079) then 2^m is a term. Indeed, 2^m has a single 1 in binary expansion and (2^m)' = m*2^(m-1) = 2^(2^k + k - 1) also has a single 1 in its binary expansion.
If p is a prime number of the form p = 4*k + 1 (A002144), then 4*p is a term. Indeed, if p is in A002144, then its binary expansion is of the form b_1_b_2...b_k01, and for 4*p it is b_1_b_2...b_k0100. Because (4*p)' = 4*(p + 1) = 4*p + 4 the binary expansion of the number 4*p + 4 is b_1_b_2....b_k0100 + 100 = b_1_b_2....b_k1000, so with the same number of digits 1.
LINKS
EXAMPLE
0' = 0, so 0 is a term.
2 = 10_2 and 2' = 1 = 1_2, so 2 is a term.
20 = 10100_2 and 20' = 24 = 11000_2, so 20 is a term.
MAPLE
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> (f-> is(f(n)=f(d(n))))(t-> add(i, i=Bits[Split](t))):
select(q, [$0..300])[]; # Alois P. Heinz, Jan 26 2022
MATHEMATICA
d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 300], Equal @@ DigitCount[{#, d[#]}, 2, 1] &] (* Amiram Eldar, Jan 25 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)]])>; [n: n in [0..265]|&+Intseq(n, 2) eq &+Intseq(Floor(f(n)), 2)];
(PARI) d(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = hammingweight(k) == hammingweight(d(k)); \\ Michel Marcus, Jan 25 2022
CROSSREFS
Sequence in context: A074677 A179997 A101756 * A352359 A226007 A257655
KEYWORD
nonn,base
AUTHOR
Marius A. Burtea, Jan 24 2022
STATUS
approved

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Last modified March 1 15:55 EST 2024. Contains 370442 sequences. (Running on oeis4.)