A390953 Odd Achilles numbers.

675, 1125, 1323, 3087, 3267, 4563, 6075, 6125, 7803, 8575, 9747, 10125, 11907, 11979, 14283, 15125, 16875, 19773, 21125, 22707, 25947, 27783, 28125, 29403, 30375, 33075, 33275, 36125, 36963, 41067, 41503, 44217, 45125, 45387, 49923, 54675, 54925, 55125, 57967, 59643

Offset

1,1

Comments

Odd powerful numbers that are not perfect powers.

Intersection of A005408 and A052486 = A062739 \ A075109 = A363217 \ A075109.

Links

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

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) - Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) - 1 = 0.0066934004068073072026... . - Amiram Eldar, Nov 29 2025

Example

Table n, a(n) for select n:
 n     a(n)
-----------------------------
 1     675 = 3^3 *  5^2
 2    1125 = 3^2 *  5^3
 3    1323 = 3^3 *  7^2
 4    3087 = 3^2 *  7^3
 5    3267 = 3^3 * 11^2
 6    4563 = 3^3 * 13^2
 7    6075 = 3^5 *  5^2
 8    6125 = 5^3 *  7^2
 9    7803 = 3^3 * 17^2
10    8575 = 5^2 *  7^3
11    9747 = 3^3 * 19^2
26   33075 = 3^3 *  5^2 * 7^2

Maple

q:= n-> (l-> min(l)>1 and igcd(l[])=1)(ifactors(n)[2][.., 2]):
select(q, [2*i-1$i=1..29822])[];  # Alois P. Heinz, Nov 28 2025

Mathematica

With[{nn = 60000}, Rest@ Union@ Flatten@ Table[If[OddQ[#] && GCD @@ FactorInteger[#][[;; , -1]] == 1, #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]

Prog

(SageMath)  # The function 'isAchilles' is defined in A390952.
def isA390953(n): return not 2.divides(n) and isAchilles(n)
print([n for n in range(600, 60000) if isA390953(n)]) # Peter Luschny, Nov 28 2025
(PARI) isok(k) = k > 1 && k % 2 && ispowerful(k) && !ispower(k); \\ Amiram Eldar, Nov 28 2025

Crossrefs

Cf. A001597, A001694, A005408, A052486, A062739, A075109, A126706, A286708, A363217, A390952.

Sequence in context: A264328 A395674 A379164 * A124942 A158393 A294949

Adjacent sequences: A390950 A390951 A390952 * A390954 A390955 A390956

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 26 2025

Status

approved