A376271 Numbers k such that there exists at least one proper divisor that is neither squarefree nor a prime power, i.e., m is in A126706.

24, 36, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272

Offset

1,1

Comments

Numbers k such that A376514(k) > 1. A376514(k) >= 1 for all k in A126706.

Numbers k such that the cardinality of the intersection of row n of A027750 and A126706 exceeds 1.

a(n) is not in A366825, since for k in A366825, there is only one divisor that is in A126706, and that is k itself.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Intersection of A033987 and A126706, i.e., { k : bigomega(k) > omega(k) > 1, bigomega(k) > 3 }, where bigomega = A001222 and omega(k) = A001221. - Michael De Vlieger, Dec 24 2024

Example

4 is not in the sequence since 4 is a prime power, and all divisors d | k of prime power k = p^e are also prime powers.
6 is not in the sequence since 6 is squarefree, and all divisors d | k of squarefree k are also squarefree.
12 is not in the sequence since 12 is in A366825, and there is only 1 divisor in A126706, which is 12 itself.
24 is in the sequence since the intersection of A126706 and row 24 of A027750, indicated by bracketed numbers, is {1, 2, 3, 4, 6, [12, 24]}, etc.
Table listing the intersection of A126706 and row a(n) of A027750 for n <= 12:
  24: {12, 24}
  36: {12, 18, 36}
  40: {20, 40}
  48: {12, 24, 48}
  54: {18, 54}
  56: {28, 56}
  60: {12, 20, 60}
  72: {12, 18, 24, 36, 72}
  80: {20, 40, 80}
  84: {12, 28, 84}
  88: {44, 88}
  90: {18, 45, 90}

Mathematica

Select[Range[300], Function[k, DivisorSum[k, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &] > 1]]
(* Second program *)
Select[Range[300], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &] (* Michael De Vlieger, Dec 24 2024 *)

Prog

(PARI) list(lim)=my(v=List()); forfactored(k=24, lim\1, my(e=k[2][, 2]); if(#e>1 && vecmax(e)>1 && (#e>2 || vecsum(e)>3), listput(v, k[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 01 2024

Crossrefs

Cf. A007947, A027750, A033987, A126706, A366825, A376514.

Sequence in context: A102505 A112064 A090440 * A091192 A067807 A224907

Adjacent sequences: A376268 A376269 A376270 * A376272 A376273 A376274

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 28 2024

Status

approved