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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). 14
 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: A216426 A272191 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 December 3 18:47 EST 2016. Contains 278745 sequences.