A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008

Offset: 1

Author

Amiram Eldar, Jul 12 2024

Keywords

• nonn
• easy

Comments

Subsequence of A322448 and first differs from it at n = 138: A322448(138) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k composite} (1/zeta(k+1) - 1/zeta(k)) = 0.05296279266796920306... . The asymptotic density of this sequence within the nonsquarefree numbers (A013929) is d / (1 - 1/zeta(2)) = 0.13508404411123191108... .

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002808, A013661, A051903, A229099.

Complement of A074661 within A013929.

Subsequence of A322448 and A322449 \ {1}.

Similar sequences: A368714, A369937, A369938, A369939, A374589, A374590.

Programs

Maple

filter:= proc(n) local m;
  m:= max(ifactors(n)[2][..,2]);
  m > 1 and not isprime(m)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 14 2024

Mathematica

Select[Range[1200], CompositeQ[Max[FactorInteger[#][[;; , 2]]]] &]

PARI

iscomposite(n) = n > 1 && !isprime(n);
is(n) = n > 1 && iscomposite(vecmax(factor(n)[, 2]));