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Numbers n such that n = A008477(A008477(n)) and n is not A008478.
8

%I #20 Mar 29 2021 14:49:31

%S 8,9,25,32,49,121,125,128,169,200,243,288,289,343,361,392,500,529,675,

%T 841,864,961,968,972,1125,1152,1323,1331,1352,1369,1372,1568,1681,

%U 1849,1944,2000,2048,2187,2197,2209,2312,2809,2888,3087,3200,3267,3456,3481

%N Numbers n such that n = A008477(A008477(n)) and n is not A008478.

%C From _Bernard Schott_, Mar 29 2021: (Start)

%C If m is a term, then A008477(m) = q is another term and A008477(q) = m.

%C The first such pairs (m, q) in lexicographic order are (8, 9), (25, 32), (49, 128), (121, 2048), (125, 243), (169, 8192), (200, 288), (289, 131072), ...

%C If f = A008477, terms k of this sequence are precisely the ones for which the sequence k, f(k), f(f(k)), f(f(f(k))), ... is periodic with period = 2 (see 1st comment in A008477); example for k = 8, this periodic sequence is 8, 9, 8, 9, 8, 9, ...

%C Prime powers p^r, p, r primes, p <> r are terms. (End)

%H Michel Marcus, <a href="/A062307/b062307.txt">Table of n, a(n) for n = 1..4336</a>

%e 8 = 2^3, A008477(8) = 3^2 = 9 and A008477(9) = 2^3 = 8, so 8 and 9 are terms.

%e 200 = 2^3*5^2, A008477(200) = 3^2*2^5 = 288 and A008477(288) = 2^3*5^2 = 200, so 200 and 288 are terms.

%o (PARI) f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477

%o isok(m) = my(nm = f(m)); (nm != m) && (f(nm) == m); \\ _Michel Marcus_, Mar 29 2021

%Y A114130 is a subsequence.

%K easy,nonn

%O 1,1

%A _Naohiro Nomoto_, Mar 28 2002