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A052486 Achilles numbers - powerful but imperfect: writing n=product(p_i^e_i) then none of the e_i=1 (i.e., powerful(1)) but the highest common factor of the e_i>1 is 1 (so not perfect powers). 13
72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Number of terms < 10^n: 0, 1, 13, 60, 252, 916, 3158, 10553, 34561, 111891, 359340, 1148195, 3656246, 11616582, 36851965, ..., A118896(n) - A070428(n). - Robert G. Wilson v, Aug 11 2014

a(n) = (s(n))^2 * f(n), s(n) > 1, f(n) > 1, where s(n) is not a power of f(n), and f(n) is squarefree and gcd(s(n), f(n)) = f(n). - Daniel Forgues, Aug 11 2015

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Project Euler, Problem 302.

Robert Israel, log-log plot of a(n)

Eric Weisstein's World of Mathematics, Achilles Number

FORMULA

a(n) = O(n^2). - Daniel Forgues, Aug 11 2015

a(n) = O(n^2 / log log n). - Daniel Forgues, Aug 12 2015

EXAMPLE

a(3)=200 because 200=2^3*5^2, both 3 and 2 are greater than 1, and the highest common factor of 3 and 2 is 1.

Factorizations of a(1) to a(20):

    72 = 2^3  3^2,  108 = 2^2 3^3,  200 = 2^3 5^2,  288 = 2^5  3^2,

   392 = 2^3  7^2,  432 = 2^4 3^3,  500 = 2^2 5^3,  648 = 2^3  3^4,

   675 = 3^3  5^2,  800 = 2^5 5^2,  864 = 2^5 3^3,  968 = 2^3 11^2,

   972 = 2^2  3^5, 1125 = 3^2 5^3, 1152 = 2^7 3^2, 1323 = 3^3  7^2,

  1352 = 2^3 13^2, 1372 = 2^2 7^3, 1568 = 2^5 7^2, 1800 = 2^3  3^2 5^2.

Examples for a(n) = (s(n))^2 * f(n): (see above comment)

s(n) = 6,  6, 10, 12, 14, 12, 10, 18, 15, 20, 12, 22, 18, 15, 24, 21,

f(n) = 2,  3,  2,  2,  2,  3,  5,  2,  3,  2,  6,  2,  3,  5,  2,  3,

MAPLE

filter:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:

select(filter, [$1..10000]); # Robert Israel, Aug 11 2014

MATHEMATICA

achillesQ[n_] := Block[{ls = Last /@ FactorInteger@n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 5500, achillesQ@# &] (* Robert G. Wilson v, Jun 10 2010 *)

PROG

(PARI) isA052486(n)= n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n) \\ M. F. Hasler, Sep 23 2010

(PARI) is(n)=my(f=factor(n)[, 2]); n>9 && vecmin(f)>1 && gcd(f)==1 \\ Charles R Greathouse IV, Sep 18 2015

CROSSREFS

Cf. A001597, A001694, A007916.

Sequence in context: A102562 A216426 A072412 * A114128 A143610 A166987

Adjacent sequences:  A052483 A052484 A052485 * A052487 A052488 A052489

KEYWORD

nonn,more

AUTHOR

Henry Bottomley, Mar 16 2000

EXTENSIONS

Example edited by Mac Coombe (mac.coombe(AT)gmail.com), Sep 18 2010

STATUS

approved

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Last modified February 10 18:55 EST 2016. Contains 268157 sequences.