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

Author

Henry Bottomley, Jun 25 2000

Keywords

• nonn
• mult

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Henry Bottomley, Some Smarandache-type multiplicative sequences.

Crossrefs

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

Cf. A013662, A013674.

Programs

Mathematica

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

PARI

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

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