

A053166


Smallest positive integer for which n divides a(n)^4.


13



1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 4, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 4, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38
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OFFSET

1,2


COMMENTS

According to Broughan (2002, 2003, 2006), a(n) is the "upper 4th root of n". The "lower 4th root of n" is sequence A053164.  Petros Hadjicostas, Sep 15 2019


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandachetype multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105114.
Kevin A. Broughan, Relationship between the integer conductor and kth root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121136.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.


FORMULA

a(n) = n/A000190(n) = A019554(n)/(A008835(A019554(n)^2))^(1/4).
If n is 5thpowerfree (i.e., not 32, 64, 128, 243, ...) then a(n) = A007947(n).
Multiplicative with a(p^e) = p^(ceiling(e/4)).  Christian G. Bower, May 16 2005


MATHEMATICA

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


PROG

(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = ceil(f[i, 2]/4)); factorback(f); \\ Michel Marcus, Jun 09 2014


CROSSREFS

Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A015052 (outer 5th root), A015053 (outer 6th root).
Cf. A000189, A000190, A008835, A015051.
Sequence in context: A015053 A062953 A015052 * A166140 A019555 A243074
Adjacent sequences: A053163 A053164 A053165 * A053167 A053168 A053169


KEYWORD

nonn,mult


AUTHOR

Henry Bottomley, Feb 29 2000


STATUS

approved



