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

1, 4, 9, 8, 16, 32, 27, 25, 64, 128, 81, 72, 512, 1024, 108, 2048, 243, 49, 4096, 8192, 16384, 288, 729, 32768, 125, 225, 200, 131072, 262144, 2187, 524288, 1152, 1048576, 432, 2097152, 4194304, 972, 196, 8388608, 648, 33554432, 4608, 864, 67108864, 19683, 268435456

Offset: 1

Author

Bernard Schott, Mar 27 2021

Keywords

• nonn

Comments

As A008477 is not injective and terms A008477(n) are precisely the powerful numbers, this sequence lists the smallest preimage of each powerful number.

There are these three possibilities (see corresponding examples):

-> If A008477(q) = q is a fixed point in A008478 and if q = A001694(u) then a(u) = q.

-> 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;

-> 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.

Examples

			-> A008477(16) = 16 is a fixed point and 16 is the 5th powerful number, so a(5) = 16.
-> 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.
-> A008477(81) = A008477(256) = 64 that is the 11th powerful number, since 81 is the smallest preimage of 64, so a(11) = 81.
		

Crossrefs

Cf. A001694, A008477, A008478, A062307.

Programs

PARI

pwf(n) = my(k=1, nb=1); while (nb != n, k++; if (ispowerful(k), nb++)); k; \\ A001694
f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
a(n) = my(k=1, p=pwf(n)); while (f(k) != p, k++); k; \\ Michel Marcus, Mar 28 2021

Extensions

More terms from Amiram Eldar, Mar 27 2021