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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.

%I A376271 #42 Dec 24 2024 12:54:02
%S A376271 24,36,40,48,54,56,60,72,80,84,88,90,96,100,104,108,112,120,126,132,
%T A376271 135,136,140,144,150,152,156,160,162,168,176,180,184,189,192,196,198,
%U A376271 200,204,208,216,220,224,225,228,232,234,240,248,250,252,260,264,270,272
%N 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.
%C A376271 Numbers k such that A376514(k) > 1. A376514(k) >= 1 for all k in A126706.
%C A376271 Numbers k such that the cardinality of the intersection of row n of A027750 and A126706 exceeds 1.
%C A376271 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 A376271 Michael De Vlieger, <a href="/A376271/b376271.txt">Table of n, a(n) for n = 1..10000</a>
%F A376271 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 A376271 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 A376271 6 is not in the sequence since 6 is squarefree, and all divisors d | k of squarefree k are also squarefree.
%e A376271 12 is not in the sequence since 12 is in A366825, and there is only 1 divisor in A126706, which is 12 itself.
%e A376271 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 A376271 Table listing the intersection of A126706 and row a(n) of A027750 for n <= 12:
%e A376271   24: {12, 24}
%e A376271   36: {12, 18, 36}
%e A376271   40: {20, 40}
%e A376271   48: {12, 24, 48}
%e A376271   54: {18, 54}
%e A376271   56: {28, 56}
%e A376271   60: {12, 20, 60}
%e A376271   72: {12, 18, 24, 36, 72}
%e A376271   80: {20, 40, 80}
%e A376271   84: {12, 28, 84}
%e A376271   88: {44, 88}
%e A376271   90: {18, 45, 90}
%t A376271 Select[Range[300], Function[k, DivisorSum[k, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &] > 1]]
%t A376271 (* Second program *)
%t A376271 Select[Range[300], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &] (* _Michael De Vlieger_, Dec 24 2024 *)
%o A376271 (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 A376271 Cf. A007947, A027750, A033987, A126706, A366825, A376514.
%K A376271 nonn,easy
%O A376271 1,1
%A A376271 _Michael De Vlieger_, Sep 28 2024