A070875 -id:A070875 - cp's OEIS frontend

A063920 Numbers k such that k = 2*phi(k) + phi(phi(k)).

10, 14, 20, 28, 40, 56, 80, 112, 160, 224, 320, 448, 640, 896, 1280, 1792, 2560, 3584, 5120, 7168, 10240, 14336, 20480, 28672, 40960, 57344, 81920, 114688, 163840, 229376, 327680, 458752, 655360, 917504, 1310720, 1835008, 2621440, 3670016, 5242880, 7340032, 10485760

Offset: 0

Jason Earls, Aug 31 2001

Previous name was: t(n) = z(n) where t(n)= |eulerphi(n)-n| and z(n)= t(t(n)-n).

Amiram Eldar, Table of n, a(n) for n = 0..6637
Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.
Lawrence Sze, Conjecture 36 (at archive.org).
Lawrence Sze, Conjecture 36 - from OEIS - a.k.a. A063920, preprint, 2004. [cached copy]
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A070875 (the same sequence, if we omit the two initial terms).

[(12-2*(-1)^n)*2^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Feb 29 2016

CoefficientList[Series[(10 + 14 x) / (1 - 2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 29 2016 *)

t(n) = abs(eulerphi(n)-n); z(n) = t(t(n)-n);
for(n=1,113, if(t(n)==z(n),print1(n, ", ")))

G.f.: (10 + 14x)/(1 - 2x^2).
a(n) = (12-2*(-1)^n) * 2^floor(n/2). - Ralf Stephan, Jul 19 2013
Sum_{n>=0} 1/a(n) = 12/35. - Amiram Eldar, Mar 28 2022

Better name from Ivan Neretin, Feb 28 2016

A281392 Number of occurrences of "01" as a subsequence in the binary expansion of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 4, 3, 4, 1, 5, 4, 7, 3, 6, 5, 7, 2, 5, 4, 6, 3, 5, 4, 5, 1, 6, 5, 9, 4, 8, 7, 10, 3, 7, 6, 9, 5, 8, 7, 9, 2, 6, 5, 8, 4, 7, 6, 8, 3, 6, 5, 7, 4, 6, 5, 6, 1, 7, 6, 11, 5, 10, 9, 13, 4, 9, 8, 12, 7, 11, 10, 13, 3, 8, 7, 11, 6, 10, 9, 12, 5, 9, 8, 11, 7, 10, 9, 11, 2, 7, 6, 10, 5, 9, 8, 11, 4, 8, 7, 10, 6, 9, 8, 10, 3, 7, 6, 9, 5, 8, 7, 9, 4

Offset: 0

Jeffrey Shallit, Jan 21 2017

nonn

By "subsequence" we do not demand that occurrences are contiguous. Furthermore, we assume that the binary expansion of n begins with a 0, and is read starting with the most significant digit.

			For n = 5, the number of occurrences of 01 as a subsequence of 0101 is 3.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A055941, which gives the analogous sequence for the pattern "10" instead of "01".

Cf. A000079 (a(n)=1), A007283 (a(n)=2), A070875 (a(n)=3).

f:= proc(n) option remember;
  if n::even then procname(n/2)
  elif n mod 4 = 3 then - procname((n-3)/4) + 2*procname((n-1)/2)
elif n mod 8 = 1 then -procname((n+3)/4) + 2*procname((n+1)/2)
elif n mod 16 = 5 then -2*procname((n+3)/8) + 2*procname((n-1)/4) + procname((n+5)/2)
elif n mod 16 = 13 then -procname((n-13)/16) +procname((n-5)/8) + procname((n-3)/2)
fi;
end proc:
f(0):= 0: f(1):= 1: f(5):= 3:
map(f, [$0..200]); # Robert Israel, Mar 11 2020

f[n_] := f[n] = Which[
EvenQ[n], f[n/2],
Mod[n, 4] == 3, -f[(n-3)/4] + 2*f[(n-1)/2],
Mod[n, 8] == 1, -f[(n+3)/4] + 2*f[(n+1)/2],
Mod[n, 16] == 5, -2*f[(n+3)/8] + 2*f[(n-1)/4] + f[(n+5)/2],
Mod[n, 16] == 13, -f[(n-13)/16] + f[(n-5)/8] + f[(n-3)/2]];
f[0] = 0; f[1] = 1; f[5] = 3;
Map[f, Range[0, 200]] (* Jean-François Alcover, Sep 19 2022, after Robert Israel *)

Completely defined by the recurrence relations
a(2n) = a(n); a(4n+3) = -a(n) + 2a(2n+1); a(8n+1) = -a(2n+1) + 2a(4n+1); a(16n+5) = -2a(2n+1) + 2a(4n+1) + a(8n+5);
a(16n+13) = -a(n) + a(2n+1) + a(8n+5) with a(0) = 0, a(1) = 1 and a(5) = 3.

cp's OEIS Frontend

A063920 Numbers k such that k = 2*phi(k) + phi(phi(k)).

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