A376173 Numbers whose prime factorization has an odd minimum exponent that is larger than 1.

8, 27, 32, 125, 128, 216, 243, 343, 432, 512, 648, 864, 1000, 1331, 1728, 1944, 2000, 2048, 2187, 2197, 2744, 3125, 3375, 3456, 4000, 4913, 5000, 5488, 5832, 6859, 6912, 7776, 8000, 8192, 9261, 10125, 10648, 10976, 12167, 13824, 15552, 16000, 16807, 16875, 17496

Offset: 1

Author

Amiram Eldar, Sep 13 2024

Keywords

• nonn
• easy

Comments

Numbers k such that A051904(k) is odd and larger than 1.

The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is not a term of this sequence.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to powerful numbers.

Crossrefs

Subsequence of A036966.

Complement of A376172 within A001694.

Subsequences: A030078, A062838 \ {1}.

Cf. A051904.

Programs

Mathematica

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[10000]

PARI

is(k) = {my(f = factor(k), e = f[,2]); #e && (ispowerful(f) && vecmin(e) % 2);}

Formula

Sum_{n>=1} 1/a(n) = -1 + Sum_{k>=2} (-1)^k * s(k) = 0.2379998147971880759099..., where s(k) = Product_{p prime} (1 + 1/(p^k*(p-1))).