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A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
LINKS
EXAMPLE
-> 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.
PROG
(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
CROSSREFS
Sequence in context: A308289 A236630 A351381 * A359681 A257851 A133790
KEYWORD
nonn
AUTHOR
Bernard Schott, Mar 27 2021
EXTENSIONS
More terms from Amiram Eldar, Mar 27 2021
STATUS
approved

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Last modified March 28 16:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)