A370787 -id:A370787 - cp's OEIS frontend

A370785 Powerful numbers with an even number of prime factors (counted with multiplicity).

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 216, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 864, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 1944, 2025, 2116, 2209

Offset: 1

Amiram Eldar, Mar 02 2024

Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) + 1/zeta(3/2))/2 = 1.278023... .
The relative asymptotic density of this sequence within the powerful numbers is (1 + zeta(3)/(zeta(3/2)^2))/2 = 0.588069... .
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an even number of prime factors (counted with multiplicity) within the s-full numbers is larger than 1/2 when s is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Intersection of A001694 and A028260.
Complement of A370786 within A001694.
A370787 is a subsequence.
Cf. A002117, A078434, A090699.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, n == 1 || AllTrue[e, # > 1 &] && EvenQ[Total[e]]]; Select[Range[2500], q]

is(n) = {my(e = factor(n)[, 2]); n == 1 || (vecmin(e) > 1 && !(vecsum(e)%2));}

A370788 Cubefull numbers with an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

8, 27, 32, 125, 128, 243, 343, 432, 512, 648, 1331, 1728, 2000, 2048, 2187, 2197, 2592, 3125, 3888, 4913, 5000, 5488, 5832, 6859, 6912, 8000, 8192, 10125, 10368, 12167, 15552, 16807, 16875, 19208, 19683, 20000, 21296, 21952, 23328, 24389, 27000, 27648, 27783, 29791

Offset: 1

Amiram Eldar, Mar 02 2024

Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * x^(1/3) / 2 + o(x^(1/3)) where c = A362974.
The relative asymptotic density of this sequence within the cubefull numbers is 1/2.
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an odd number of prime factors (counted with multiplicity) within the s-full numbers is 1/2 when s is odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Intersection of A036966 and A026424.
Complement of A370787 within A036966.
Subsequence of A370786.
Cf. A362974.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 2 &] && OddQ[Total[e]]]; Select[Range[30000], q]

is(n) = {my(e = factor(n)[, 2]); n > 1 && vecmin(e) > 2 && vecsum(e)%2;}

cp's OEIS Frontend

A370785 Powerful numbers with an even number of prime factors (counted with multiplicity).

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