A386434 Achilles numbers k such that A053669(k) < A006530(k).
200, 392, 500, 675, 800, 968, 1125, 1323, 1352, 1372, 1568, 2000, 2312, 2888, 3087, 3200, 3267, 3528, 3872, 4000, 4232, 4563, 5000, 5292, 5324, 5408, 5488, 6075, 6125, 6272, 6728, 7688, 7803, 8575, 8712, 8788, 9248, 9747, 9800, 10125, 10584, 10952, 10976, 11552
Offset: 1
Keywords
Examples
Let s = A052486, q = A053669, and gpf = A006530. Table of n, a(n), and q(a(n)) for n = 1..12: n a(n) q(a(n)) -------------------------------- 1 200 = 2^3 * 5^2 3 2 392 = 2^3 * 7^2 3 3 500 = 2^2 * 5^3 3 4 675 = 3^3 * 5^2 2 5 800 = 2^5 * 5^2 3 6 968 = 2^3 * 11^2 3 7 1125 = 3^2 * 5^3 2 8 1323 = 3^3 * 7^2 2 9 1352 = 2^3 * 13^2 3 10 1372 = 2^2 * 7^3 3 11 1568 = 2^5 * 7^2 3 12 2000 = 2^4 * 5^3 3 The number 12 is not a term since it is not powerful (i.e., not in A001694). The number 36, though powerful, is not a term since it is a perfect square. s(1) = 72 is not in the sequence since q(72) > gpf(72), i.e., 5 > 3. s(2) = 108 is not in the sequence since q(108) > gpf(108), i.e., 5 > 3. a(1) = s(3) = 200 because q(200) < gpf(200), i.e., 3 < 5. a(2) = s(4) = 392 because q(392) < gpf(392), i.e., 3 < 7, etc.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *) 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]] ] ]