A015053 Smallest positive integer for which n divides a(n)^6.

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, 2, 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, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset

1,2

Comments

Differs from A007947 as follows: A007947(128)=2, a(128)=4; A007947(256)=2, a(256)=4; A007947(384)=6, a(384)=12; A007947(512)=2, a(512)=4; A007947(640)=10, a(640)=20, etc. - R. J. Mathar, Oct 28 2008

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.

Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.

Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.

Henry Ibstedt, Surfing on the Ocean of Numbers, Erhus Univ. Press, Vail, 1997.

Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Formula

Multiplicative with a(p^e) = p^ceiling(e/6). - Christian G. Bower, May 16 2005

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(11)/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 + 1/p^9 - 1/p^10) = 0.3522558764... . - Amiram Eldar, Oct 27 2022

Mathematica

spi[n_]:=Module[{k=1}, While[PowerMod[k, 6, n]!=0, k++]; k]; Array[spi, 80] (* Harvey P. Dale, Feb 29 2020 *)
f[p_, e_] := p^Ceiling[e/6]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

Prog

(PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = ceil(f[i, 2]/6)); factorback(f); \\ Michel Marcus, Feb 15 2015
(Python)
from math import prod
from sympy import factorint
def A015053(n): return prod(p**((r:=divmod(e, 6))[0]+bool(r[1])) for p, e in factorint(n).items()) # Chai Wah Wu, Sep 17 2025

Crossrefs

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), A015052 (5th outer root).

Cf. A013669.

Sequence in context: A346486 A088835 A007947 * A062953 A347230 A015052

Adjacent sequences: A015050 A015051 A015052 * A015054 A015055 A015056

Keyword

nonn,mult,easy

Author

R. Muller (Research37(AT)aol.com)

Extensions

Corrected by David W. Wilson, Jun 04 2002

Status

approved