A376271 - OEIS

%I #42 Dec 24 2024 12:54:02

%S 24,36,40,48,54,56,60,72,80,84,88,90,96,100,104,108,112,120,126,132,

%T 135,136,140,144,150,152,156,160,162,168,176,180,184,189,192,196,198,

%U 200,204,208,216,220,224,225,228,232,234,240,248,250,252,260,264,270,272

%N 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.

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

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

%C 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.

%H Michael De Vlieger, <a href="/A376271/b376271.txt">Table of n, a(n) for n = 1..10000</a>

%F 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

%e 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.

%e 6 is not in the sequence since 6 is squarefree, and all divisors d | k of squarefree k are also squarefree.

%e 12 is not in the sequence since 12 is in A366825, and there is only 1 divisor in A126706, which is 12 itself.

%e 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.

%e Table listing the intersection of A126706 and row a(n) of A027750 for n <= 12:

%e   24: {12, 24}

%e   36: {12, 18, 36}

%e   40: {20, 40}

%e   48: {12, 24, 48}

%e   54: {18, 54}

%e   56: {28, 56}

%e   60: {12, 20, 60}

%e   72: {12, 18, 24, 36, 72}

%e   80: {20, 40, 80}

%e   84: {12, 28, 84}

%e   88: {44, 88}

%e   90: {18, 45, 90}

%t Select[Range[300], Function[k, DivisorSum[k, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &] > 1]]

%t (* Second program *)

%t Select[Range[300], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &] (* _Michael De Vlieger_, Dec 24 2024 *)

%o (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

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

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Sep 28 2024