A377590 Numbers k neither squarefree nor prime powers such that there exist no numbers m such that rad(m) | k and Omega(m) > Omega(k), where rad = A007947 and Omega = A001222.

12, 24, 45, 48, 63, 75, 96, 135, 175, 189, 192, 225, 245, 275, 325, 384, 405, 425, 475, 539, 567, 575, 605, 637, 675, 768, 833, 847, 875, 931, 1127, 1183, 1215, 1225, 1375, 1421, 1519, 1536, 1573, 1625, 1701, 1715, 1813, 1859, 1925, 2009, 2023, 2025, 2057, 2107

Offset: 1

Author

Michael De Vlieger, Nov 02 2024

Keywords

• nonn
• easy

Comments

This sequence contains numbers k in A126706 for which A376846(k) = 0; A376846(k) = 0 for prime powers k or squarefree numbers k (i.e., k in A303554).

It is sufficient to determine floor(log k / log p) <= Omega(k) for p = lpf(k) = A020639(k).

Sequence contains numbers k of the form 2^j*3, j > 1, i.e., A007283 \ {3, 6} is a proper subset of this sequence, since 2^(j+1) < 2^j*3 and j+1 = Omega(2^j*3).

The numbers k that remain in the sequence ({a(n)} \ A007283) are odd, that is, in A360769. For k = 2^j*p, prime p > 3, we have j+floor(log_2 p) > j+1, since log_2 p > 2, therefore we see m = 2^(j+floor(log_2 p)) < 2^j*p, with Omega(m) > Omega(k).

Examples

			12 is in the sequence since 2^3 < 12, and Omega(2^3) = Omega(12) = 3.
20 is not in the sequence since 2^4 < 20 and Omega(2^4) = 4, but Omega(20) = 3.
45 is in the sequence since 3^3 < 45, and Omega(3^3) = Omega(45) = 3.
375 = 3*5^3 is not in the sequence since 3^5 < 375 and Omega(3^5) = 5, but Omega(345) = 4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024.

Crossrefs

Cf. A001222, A007283, A007947, A020639, A126706, A303554, A360769.

Programs

Mathematica

Select[Select[Range[4000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{n, k}, NoneTrue[FactorInteger[n][[All, 1]], Floor@ Log[#, n] > k &]] @@ {#, PrimeOmega[#]} &]