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A386434 Achilles numbers k such that A053669(k) < A006530(k).

%I A386434 #6 Jul 21 2025 13:15:28
%S A386434 200,392,500,675,800,968,1125,1323,1352,1372,1568,2000,2312,2888,3087,
%T A386434 3200,3267,3528,3872,4000,4232,4563,5000,5292,5324,5408,5488,6075,
%U A386434 6125,6272,6728,7688,7803,8575,8712,8788,9248,9747,9800,10125,10584,10952,10976,11552
%N A386434 Achilles numbers k such that A053669(k) < A006530(k).
%H A386434 Michael De Vlieger, <a href="/A386434/b386434.txt">Table of n, a(n) for n = 1..10000</a>
%F A386434 Intersection of A052486 and A080259 = A052486 \ A377854.
%e A386434 Let s = A052486, q = A053669, and gpf = A006530.
%e A386434 Table of n, a(n), and q(a(n)) for n = 1..12:
%e A386434  n    a(n)               q(a(n))
%e A386434 --------------------------------
%e A386434  1    200 = 2^3 * 5^2    3
%e A386434  2    392 = 2^3 * 7^2    3
%e A386434  3    500 = 2^2 * 5^3    3
%e A386434  4    675 = 3^3 * 5^2    2
%e A386434  5    800 = 2^5 * 5^2    3
%e A386434  6    968 = 2^3 * 11^2   3
%e A386434  7   1125 = 3^2 * 5^3    2
%e A386434  8   1323 = 3^3 * 7^2    2
%e A386434  9   1352 = 2^3 * 13^2   3
%e A386434 10   1372 = 2^2 * 7^3    3
%e A386434 11   1568 = 2^5 * 7^2    3
%e A386434 12   2000 = 2^4 * 5^3    3
%e A386434 The number 12 is not a term since it is not powerful (i.e., not in A001694).
%e A386434 The number 36, though powerful, is not a term since it is a perfect square.
%e A386434 s(1) = 72 is not in the sequence since q(72) > gpf(72), i.e., 5 > 3.
%e A386434 s(2) = 108 is not in the sequence since q(108) > gpf(108), i.e., 5 > 3.
%e A386434 a(1) = s(3) = 200 because q(200) < gpf(200), i.e., 3 < 5.
%e A386434 a(2) = s(4) = 392 because q(392) < gpf(392), i.e., 3 < 7, etc.
%t A386434 (* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
%t A386434 nn = 6; mm = Times @@ Prime@ Range[nn]; Complement[Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], And[Length[#2] > 1, GCD @@ #2 == 1] & @@ {#, FactorInteger[#][[;; , -1]]} &], Union@ Flatten[f[nn][[3 ;; -1, 2 ;; -1]] ] ]
%Y A386434 Cf. A001597, A001694, A006530, A052486, A053669, A052486, A080259, A377854.
%K A386434 nonn
%O A386434 1,1
%A A386434 _Michael De Vlieger_, Jul 21 2025