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A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694).

%I A342551 #46 Mar 31 2023 09:18:28
%S A342551 1,4,9,8,16,32,27,25,64,128,81,72,512,1024,108,2048,243,49,4096,8192,
%T A342551 16384,288,729,32768,125,225,200,131072,262144,2187,524288,1152,
%U A342551 1048576,432,2097152,4194304,972,196,8388608,648,33554432,4608,864,67108864,19683,268435456
%N A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694).
%C A342551 As A008477 is not injective and terms A008477(n) are precisely the powerful numbers, this sequence lists the smallest preimage of each powerful number.
%C A342551 There are these three possibilities (see corresponding examples):
%C A342551 -> If A008477(q) = q is a fixed point in A008478 and if q = A001694(u) then a(u) = q.
%C A342551 -> If k and m are in A062307 and satisfy A008477(k) = m  and A008477(m) = k, if m = A001694(s) and k = A001694(t), then a(t) = m and a(s) = k;
%C A342551 -> If A008477(j) = v where v is a powerful number not in {A008478 U A062307} and j is the smallest preimage of v with v = A001694(z) then a(z) = j.
%e A342551 -> A008477(16) = 16 is a fixed point and 16 is the 5th powerful number, so a(5) = 16.
%e A342551 -> 25 and 32 are in A062307 and satisfy A008477(25) = 32 and A008477(32) = 25, as 25 = A001694(6) and 32 = A001694(8), so a(6) = 32 and a(8) = 25.
%e A342551 -> A008477(81) = A008477(256) = 64 that is the 11th powerful number, since 81 is the smallest preimage of 64, so a(11) = 81.
%o A342551 (PARI) pwf(n) = my(k=1, nb=1); while (nb != n, k++; if (ispowerful(k), nb++)); k; \\ A001694
%o A342551 f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
%o A342551 a(n) = my(k=1, p=pwf(n)); while (f(k) != p, k++); k; \\ _Michel Marcus_, Mar 28 2021
%Y A342551 Cf. A001694, A008477, A008478, A062307.
%K A342551 nonn
%O A342551 1,2
%A A342551 _Bernard Schott_, Mar 27 2021
%E A342551 More terms from _Amiram Eldar_, Mar 27 2021