A056553 Smallest 4th-power divisible by n divided by largest 4th-power which divides n.

1, 16, 81, 16, 625, 1296, 2401, 16, 81, 10000, 14641, 1296, 28561, 38416, 50625, 1, 83521, 1296, 130321, 10000, 194481, 234256, 279841, 1296, 625, 456976, 81, 38416, 707281, 810000, 923521, 16, 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) = A053167(n)/A008835(n) = A056556(n)*A053165(n) = A056554(n)^4.

From Amiram Eldar, Oct 27 2022: (Start)

Multiplicative with a(p^e) = 1 if e is divisible by 4, and a(p^e) = p^4 otherwise.

Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(20)/(5*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.123026157003... . (End)

Example

a(64) = 16 because smallest 4th power divisible by 64 is 256 and largest 4th power which divides 64 is 16 and 256/16 = 16.

Mathematica

f[p_, e_] := p^If[Divisible[e, 4], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^4; Array[a, 100] (* Amiram Eldar, Aug 29 2019*)

Prog

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

Crossrefs

Cf. A000190, A008835, A053164, A053165, A053166, A053167, A056551, A056554, A056555.

Cf. A013662, A013678.

Sequence in context: A334660 A088379 A005065 * A053167 A187363 A361266

Adjacent sequences: A056550 A056551 A056552 * A056554 A056555 A056556

Keyword

nonn,mult

Author

Henry Bottomley, Jun 25 2000

Status

approved