A390952 - OEIS

A390952

Even Achilles numbers.

72, 108, 200, 288, 392, 432, 500, 648, 800, 864, 968, 972, 1152, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3200, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6272, 6728, 6912, 7200, 7688, 8712, 8748, 8788

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OFFSET

1,1

COMMENTS

Even powerful numbers that are not perfect powers.

Intersection of A005843 and A052486 = A390010 \ A075090 = A363216 \ A075090.

Sequence has significant early agreement with A378859, however, a(19) = 2312 is not in A378859, since A000203(2312) = 4605 < 2*2312, and A378859(88) = 33075, since A000203(33075) = 70680 > 2*33075.

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) = zeta(2)*zeta(3)/(3*zeta(6)) + Sum_{k>=2} mu(k)*zeta(k)/2^k = 0.0624386680090066618736... . - Amiram Eldar, Nov 29 2025

EXAMPLE

Table n, a(n) for select n:

n a(n)

----------------------------

1 72 = 2^3 * 3^2

2 108 = 2^2 * 3^3

3 200 = 2^3 * 5^2

4 288 = 2^5 * 3^2

5 392 = 2^3 * 7^2

6 432 = 2^4 * 3^3

7 500 = 2^2 * 5^3

8 648 = 2^3 * 3^4

9 800 = 2^5 * 5^2

10 864 = 2^5 * 3^3

11 968 = 2^3 * 11^2

17 1800 = 2^3 * 3^2 * 5^2

MAPLE

q:= n-> (l-> min(l)>1 and igcd(l[])=1)(ifactors(n)[2][.., 2]):

select(q, [2*i$i=1..4500])[]; # Alois P. Heinz, Nov 28 2025

MATHEMATICA

With[{nn = 9000}, Union@ Flatten@ Table[If[EvenQ[#] && GCD @@ FactorInteger[#][[;; , -1]] == 1, #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]

PROG

(SageMath)

def isAchilles(n: int) -> bool:

factors = factor(n)

exps = [e for (_, e) in factors]

min_e = min(exps)

g = reduce(gcd, exps)

return g == 1 and min_e > 1

def isA390952(n): return 2.divides(n) and isAchilles(n)

print([n for n in range(1, 9001) if isA390952(n)]) # Peter Luschny, Nov 28 2025

(PARI) isok(k) = !(k % 2) && ispowerful(k) && !ispower(k); \\ Amiram Eldar, Nov 28 2025

CROSSREFS