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 A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power. 20
 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, 6075 (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 Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) Project Euler, Problem 302: Strong Achilles Numbers. Robert Israel, log-log plot of a(n). Eric Weisstein's World of Mathematics, Achilles Number. OEIS Wiki, Achilles numbers. FORMULA a(n) = O(n^2). - Daniel Forgues, Aug 11 2015 a(n) = O(n^2 / log log n). - Daniel Forgues, Aug 12 2015 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} mu(k)*(1-zeta(k)) - 1 = A082695 - A072102 - 1 = 0.06913206841581433836... - Amiram Eldar, Oct 14 2020 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) is(n)=my(f=factor(n)[, 2]); n>9 && vecmin(f)>1 && gcd(f)==1 \\ Charles R Greathouse IV, Sep 18 2015, replacing code by M. F. Hasler, Sep 23 2010 (Python) from math import gcd from itertools import count, islice from sympy import factorint def A052486_gen(startvalue=1): # generator of terms >= startvalue return (n for n in count(max(startvalue, 1)) if (lambda x: all(e > 1 for e in x) and gcd(*x) == 1)(factorint(n).values())) A052486_list = list(islice(A052486_gen(), 20)) # Chai Wah Wu, Feb 19 2022 CROSSREFS Cf. A001597, A001694, A007916, A072102, A082695. Sequence in context: A307758 A272191 A072412 * A114128 A143610 A166987 Adjacent sequences: A052483 A052484 A052485 * A052487 A052488 A052489 KEYWORD nonn AUTHOR Henry Bottomley, Mar 16 2000 EXTENSIONS Example edited by Mac Coombe (mac.coombe(AT)gmail.com), Sep 18 2010 Name edited by M. F. Hasler, Jul 17 2019 STATUS approved

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Last modified July 23 16:21 EDT 2024. Contains 374552 sequences. (Running on oeis4.)