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A032452
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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.
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5
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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
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OFFSET
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1,3
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COMMENTS
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Original definition was "Iterates phi, phi, sigma, phi, phi, sigma, ...".
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LINKS
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EXAMPLE
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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).
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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Ursula Gagelmann (gagelmann(AT)altavista.net), Apr 07 1998
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EXTENSIONS
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STATUS
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approved
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