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

1,2

Links

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

Henry Bottomley, Some Smarandache-type multiplicative sequences.

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

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*)

Prog

(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

Crossrefs

Cf. A000190, A007913, A008835.

Cf. A013662, A013674.

Sequence in context: A088379 A005065 A056553 * A187363 A361266 A110892

Adjacent sequences: A053164 A053165 A053166 * A053168 A053169 A053170

Keyword

nonn,easy,mult

Author

Henry Bottomley, Feb 29 2000

Status

approved