A056555 Smallest number k (k>0) such that n*k is a perfect 4th power.

1, 8, 27, 4, 125, 216, 343, 2, 9, 1000, 1331, 108, 2197, 2744, 3375, 1, 4913, 72, 6859, 500, 9261, 10648, 12167, 54, 25, 17576, 3, 1372, 24389, 27000, 29791, 8, 35937, 39304, 42875, 36, 50653, 54872, 59319, 250, 68921, 74088, 79507, 5324, 1125

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

Formula

a(n) = A053167(n)/n = n^3/A000190(n)^4 = A056553(n)/A053165(n).

Multiplicative with a(p^e) = p^((4 - e) mod 4). - Amiram Eldar, Sep 08 2020

Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(16)/(4*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.1537848996... . - Amiram Eldar, Oct 27 2022

Example

a(64) = 4 because the smallest 4th power divisible by 64 is 256 and 64*4 = 256.

Mathematica

f[p_, e_] := p^Mod[4 - e, 4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 08 2020 *)

Prog

(PARI) a(n, f=factor(n))=f[, 2]=-f[, 2]%4; factorback(f) \\ Charles R Greathouse IV, Apr 24 2020

Crossrefs

Cf. A000190, A008835, A048798, A053164, A053165, A053166, A053167, A056554.

Cf. A013662, A013674.

Sequence in context: A070492 A070711 A282462 * A070491 A070490 A088378

Adjacent sequences: A056552 A056553 A056554 * A056556 A056557 A056558

Keyword

nonn,mult

Author

Henry Bottomley, Jun 25 2000

Status

approved