A032452 Irregular triangle read by rows: row n >= 1 contains the sequence p(n), s(p(n)), p(s(p(n))), p(p(s(p(n)))), s(p(p(s(p(n))))), ..., repeatedly applying (p,s,p) to n, where p = phi (A000010), s = sigma = (A000203), stopping after the first 1 is reached. If 1 is never reached, row n contains -1.

1, 1, 2, 3, 2, 1, 2, 3, 2, 1, 4, 7, 6, 2, 3, 2, 1, 2, 3, 2, 1, 6, 12, 4, 2, 3, 2, 1, 4, 7, 6, 2, 3, 2, 1, 6, 12, 4, 2, 3, 2, 1, 4, 7, 6, 2, 3, 2, 1, 10, 18, 6, 2, 3, 2, 1, 4, 7, 6, 2, 3, 2, 1, 12, 28, 12, 4, 7, 6, 2, 3, 2, 1, 6, 12, 4, 2, 3, 2, 1

Offset

1,3

Comments

Original definition was "Iterates phi, phi, sigma, phi, phi, sigma, ...".

Links

Alois P. Heinz, Rows n = 1..1000, flattened

L. Alaoglu and P. Erdős, A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), 881-882.

Example

Triangle begins:
  1,
  1,
  2,  3, 2, 1,
  2,  3, 2, 1,
  4,  7, 6, 2, 3, 2, 1,
  2,  3, 2, 1,
  6, 12, 4, 2, 3, 2, 1,
  4,  7, 6, 2, 3, 2, 1,
  6, 12, 4, 2, 3, 2, 1,
  4,  7, 6, 2, 3, 2, 1,
  ...
For row n=5, for example, we get (5 -> ) phi(5) = 4 -> sigma(4) = 7 -> phi(7) = 6 -> phi(6) = 2 -> sigma(2) = 3 -> phi(3) = 2 -> phi(2) = 1 (stop).

Maple

with(numtheory):
T:= proc(n) local l, m; l:= [][]; m:= n;
      do m:=   phi(m); l:= l, m; if m=1 then break fi;
         m:= sigma(m); l:= l, m; if m=1 then break fi;
         m:=   phi(m); l:= l, m; if m=1 then break fi
      od; l
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, May 19 2020

Mathematica

f[n_, i_] := If[Mod[i, 3] == 2, DivisorSigma[1, n], EulerPhi[n]]; g[n_] := Module[{i = 1, k = n, s = {}}, While[k > 1 || i == 1, k = f[k, i++]; AppendTo[s, k]; ]; s]; Array[g, 15] // Flatten (* Amiram Eldar, May 10 2020 *)

Prog

(Sage)
N = 10
n = 1
seq = []
while n < N:
      a = euler_phi(n)
      seq.append(a)
      i = 0
      while a != 1:
          if i%3 == 0:
              a = sigma(a)
          else:
              a = euler_phi(a)
          seq.append(a)
          i += 1
      n += 1
# John Machacek, May 08 2020

Crossrefs

See A334523, A334725 for other versions.

Cf. A000010, A000203, A334685, A334686.

Sequence in context: A170823 A068073 A324504 * A084199 A277745 A353497

Adjacent sequences: A032449 A032450 A032451 * A032453 A032454 A032455

Keyword

nonn,tabf

Author

Ursula Gagelmann (gagelmann(AT)altavista.net), Apr 07 1998

Extensions

Entry revised by N. J. A. Sloane, May 09 2020. Thanks to Amiram Eldar and John Machacek for reconstructing the lost definition of this sequence.

Status

approved