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

1,1

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Intersection of A052486 and A080259 = A052486 \ A377854.

Example

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.

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]] ] ]

Crossrefs

Cf. A001597, A001694, A006530, A052486, A053669, A052486, A080259, A377854.

Sequence in context: A078492 A166256 A252059 * A004966 A117412 A109632

Adjacent sequences: A386431 A386432 A386433 * A386435 A386436 A386437

Keyword

nonn

Author

Michael De Vlieger, Jul 21 2025

Status

approved