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

6, 9, 10, 14, 18, 20, 21, 22, 24, 25, 33, 34, 35, 38, 40, 42, 44, 48, 49, 52, 62, 65, 66, 68, 69, 70, 76, 80, 84, 88, 91, 93, 94, 96, 100, 104, 110, 115, 117, 118, 121, 132, 133, 134, 138, 140, 143, 144, 145, 148, 152, 155, 158, 164, 174, 182, 185, 186, 188, 192

Offset

1,1

Comments

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.

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.

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.

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.

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.

Links

Table of n, a(n) for n=1..60.

Example

6 = 110_2 has two 1's, 6' = 5 = 101_2 has two 1's, so 6 is a term.
9 = 101_2 has two 1's, 9' = 6 = 110_2 has two 1's, so 9 is a term.
10 = 1010_2 has two 1's, 10' = 7 = 111_2 has three 1's, so 10 is a term.
18 = 10010_2 has two 1's, 18' = 21 = 10101_2 has three 1's, so 18 is a term.

Mathematica

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 *)

Prog

(Magma) fp:=func<n|IsPrime(Multiplicity(Intseq(n, 2), 1)) >; 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 [1..200]| fp(n) and fp(Floor(f(n)))];

Crossrefs

Cf. A000120, A003415, A052294, A057733, A081092, A092506, A104070, A123250.

Sequence in context: A359371 A337287 A145311 * A328244 A037198 A359277

Adjacent sequences: A363461 A363462 A363463 * A363465 A363466 A363467

Keyword

nonn,base

Author

Marius A. Burtea, Jul 08 2023

Status

approved