A053167 Smallest 4th power divisible by n.

1, 16, 81, 16, 625, 1296, 2401, 16, 81, 10000, 14641, 1296, 28561, 38416, 50625, 16, 83521, 1296, 130321, 10000, 194481, 234256, 279841, 1296, 625, 456976, 81, 38416, 707281, 810000, 923521, 256, 1185921, 1336336, 1500625, 1296, 1874161, 2085136

Offset: 1

Author

Henry Bottomley, Feb 29 2000

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000190, A007913, A008835.

Cf. A013662, A013674.

Programs

Mathematica

f[p_, e_] := p^(e + Mod[4 - Mod[e, 4], 4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019*)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + (4-f[i,2])%4));} \\ Amiram Eldar, Oct 27 2022

Formula

a(n) = (n/A000190(n))^4 = (n*A007913(n))^2/A008835(n*A007913(n)).

From Amiram Eldar, Jul 29 2022: (Start)

Multiplicative with a(p^e) = p^(e + ((4-e) mod 4)).

Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^4+3)/(p^4-1)) = 1.341459051107600424... . (End)

Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(16)/(5*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.1230279197... . - Amiram Eldar, Oct 27 2022