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
Examples
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}
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
-
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 *) -
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
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
Comments