%I #43 Oct 29 2022 04:43:19

%S 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,

%T 29,30,31,2,33,34,35,6,37,38,39,10,41,42,43,22,15,46,47,6,7,10,51,26,

%U 53,6,55,14,57,58,59,30,61,62,21,4,65,66,67,34,69,70,71,6,73,74,15,38,77,78

%N a(n) is the smallest positive integer m such that m^5 is divisible by n.

%C A multiplicative companion function n/a(n) = 1,1,1,2,1,1,1,4,3,1,1,2,1,1,1,8,1,... can be defined using the 5th instead of the 4th power in A000190, which differs from A000190 and also from A003557. - _R. J. Mathar_, Jul 14 2012

%H Amiram Eldar, <a href="/A015052/b015052.txt">Table of n, a(n) for n = 1..10000</a>

%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.

%H Kevin A. Broughan, <a href="http://www.math.waikato.ac.nz/~kab/papers/div4.pdf">Restricted divisor sums</a>, Acta Arithmetica, 101(2) (2002), 105-114.

%H Kevin A. Broughan, <a href="https://doi.org/10.4064/aa101-2-2">Restricted divisor sums</a>, Acta Arithmetica, 101(2) (2002), 105-114.

%H Kevin A. Broughan, <a href="http://ijpam.eu/contents/2003-5-3/2/2.pdf">Relationship between the integer conductor and k-th root functions</a>, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.

%H Kevin A. Broughan, <a href="https://www.thebookshelf.auckland.ac.nz/document.php?wid=2937">Relaxations of the ABC Conjecture using integer k'th roots</a>, New Zealand J. Math. 35(2) (2006), 121-136.

%H Henry Ibstedt, <a href="http://www.gallup.unm.edu/~smarandache/Ibstedt-surfing.pdf">Surfing on the Ocean of Numbers</a>, Erhus Univ. Press, Vail, 1997.

%H Florentin Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/CP2.pdf">Collected Papers, Vol. II</a>, Tempus Publ. Hse, Bucharest, 1996.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheCeilFunction.html">Smarandache Ceil Function</a>.

%F Multiplicative with a(p^e) = p^(ceiling(e/5)). - _Christian G. Bower_, May 16 2005

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(9)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6 + 1/p^7 - 1/p^8) = 0.3523622369... . - _Amiram Eldar_, Oct 27 2022

%t f[p_, e_] := p^Ceiling[e/5]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2020 *)

%o (PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = ceil(f[i,2]/5)); factorback(f); \\ _Michel Marcus_, Feb 15 2015

%Y Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015053 (outer 6th root).

%Y Cf. A013667.

%K nonn,mult,easy

%O 1,2

%A R. Muller

%E Corrected by _David W. Wilson_, Jun 04 2002

%E Name reworded by _Jon E. Schoenfield_, Oct 28 2022