A123712 -id:A123712 - cp's OEIS frontend

A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Michael De Vlieger, Oct 22 2024

Also, numbers k such that there exists a pair of necessarily composite divisors {d, k/d}, d < k/d, with quality Q, i.e., gcd(d, k/d) > 1 but there exists a prime p | d that does not divide k/d, and also a prime q | k/d that does not divide d.
A178212 is a proper subset.
This sequence is distinct from A123712 since 420 is here.
This sequence is distinct from A182855 since 360 is here.

			a(1) = 60 = 2^2 * 3 * 5, the smallest number such that bigomega(60) > omega(60) > 2. Bigomega(60) = 4, omega(60) = 3.
72 is not in the sequence because it is the product of 2 distinct prime factors.
a(2) = 84 = 2^2 * 3 * 7, since bigomega(84) = 4, omega(84) = 3.
a(3) = 90 = 2 * 3^2 * 5, since bigomega(90) = 4, omega(90) = 3.
a(4) = 120 = 2^3 * 3 * 5, since bigomega(120) = 5, omega(120) = 3.
210 is not in the sequence because it is squarefree.
a(35) = 360 = 2^3 * 3^2 * 5 since bigomega(360) = 6, omega(360) = 3.
a(43) = 420 = 2^2 * 3 * 5 * 7 since bigomega(420) = 5, omega(420) = 4, etc.
.
Table showing pairs of factors of a(n) for select n, such that the pair possesses quality Q (see comments).
    n    a(n)   pair of factors with quality Q.
  -------------------------------------------------------------------
    1     60    6 X 10;
    2     84    6 X 14;
    3     90    6 X 15;
    4    120    6 X 20,  10 X 12;
    5    126    6 X 21;
    6    132    6 X 22;
    7    140   10 X 14;
    8    150   10 X 15;
   17    240    6 X 40,  10 X 24, 12 X 20;
   51    480    6 X 80,  10 X 48, 12 X 40, 20 X 24;
  117    840    6 X 140, 10 X 84, 12 X 70, 14 X 60, 20 X 42, 28 X 30.

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

Cf. A001221, A001222, A013929, A024619, A126706, A178212.

Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]

{a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.

A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 4, 4, 6, 3, 8, 3, 6, 7, 4, 3, 8, 3, 8, 7, 6, 3, 8, 4, 6, 4, 8, 3, 11, 3, 4, 7, 6, 7, 8, 3, 6, 7, 8, 3, 11, 3, 8, 8, 6, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 6, 3, 16, 3, 6, 8, 4, 7, 12, 3, 8, 7, 14, 3, 8, 3, 6, 8, 8

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k]. a(n) = 4 when n is in A123710. a(n) = 8 when n is in A123711. a(n) = 16 when n is in A123712.

			a(n) = 3 when n is an odd prime.
a(n) = 7 when n is the product of two different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]
a(n) = 15 when n is the product of three different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]

M. F. Hasler, Table of n, a(n) for n = 1..500
Peter Luschny, Re: Is the A123706 triangle an extension of the Moebius function?, seqcomp list, Feb 12 2012

Cf. A123706, A123707, A123708, A123710, A123711, A123712; A010766.

Moebius[i_,j_]:=If[Divisible[i,j], MoebiusMu[i/j],0];
A123709[n_]:=Length[Select[Table[Moebius[n,j]-Moebius[n,j+1],{j,1,n}],#!=0&]];
Array[A123709, 500] (* Enrique Pérez Herrero, Feb 13 2012 *)

{a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1); sum(k=1,n,if(M[n,k]==0,0,1))}

A123709(n)=#select((matrix(n, n, r, c, r\c)^-1)[n,],x->x)  \\ M. F. Hasler, Feb 12 2012

A123709(n)={ my(t=moebius(n)); sum(k=2,n, t+0 != t=if(n%k,0,moebius(n\k)))+1}  /* the "t+0 != ..." is required because of a bug in PARI versions <= 2.4.2, maybe beyond, which seems to be fixed in v. 2.5.1 */ \\ M. F. Hasler, Feb 13 2012

a(n) = 2^(m+1) - 1 when n is the product of m distinct odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
For any k>1, a(n)=2^k if, and only if, n is a nonsquarefree number with A001221(n) = k-1 (= omega(n), number of distinct prime factors), with the only exception of a(n=6)=2^2. - M. F. Hasler, Feb 12 2012
A123709(n) = 1 + #{ k in 1..n-1 | Moebius(n,k+1) <> Moebius(n,k) }, where Moebius(n,k)={moebius(n/k) if n=0 (mod k), 0 else}, cf. link to message by P. Luschny. - M. F. Hasler, Feb 13 2012

A178212 Nonsquarefree numbers divisible by exactly three distinct primes.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Reinhard Zumkeller, May 24 2010

nonn

			60 is in the sequence because it is not squarefree and it is divisible by three distinct primes: 2, 3, 5.
72 is not in the sequence, because although it is not squarefree, it is divisible by only two distinct primes: 2 and 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A subsequence of A033987.
A085987 is a subsequence.
Cf. A007304, A200511, A200521.
Cf. A000005, A001221, A001222, A013929, A085987, A123712.
Cf. A027748, A124010.
Subsequence of A375055, which differs starting at a(43) = 440 > A375055(43) = 420.

a178212 n = a178212_list !! (n-1)
a178212_list = filter f [1..] where
   f x = length (a027748_row x) == 3 && any (> 1) (a124010_row x)
-- Reinhard Zumkeller, Apr 03 2015

nsD3Q[n_] := Block[{fi = FactorInteger@ n}, Length@ fi == 3 && Plus @@ Last /@ fi > 3]; Select[ Range@ 494, nsD3Q] (* Robert G. Wilson v, Feb 09 2012 *)
Select[Range[500], PrimeNu[#] == 3 && PrimeOmega[#] > 3 &] (* Alonso del Arte, Mar 23 2015, based on a comment from Robert G. Wilson v, Feb 09 2012; requires Mathematica 7.0+ *)

is_A178212(n)={ omega(n)==3 & bigomega(n)>3 }
for(n=1,999,is_A178212(n) & print1(n",")) \\ M. F. Hasler, Feb 09 2012

A001221(a(n)) = 3; A001222(a(n)) > 3; A000005(n) >= 12;

a(n) = A123712(n) for n <= 52, possibly more.

A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224, 225

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

It appears that this equals A200511, numbers of the form p^k q^m with k,m >= 1, k+m > 2 and p, q prime. - M. F. Hasler, Feb 12 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A123706, A123709, A123710, A123712; A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] :=
Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[500], A123709[#] == 8 &] (* G. C. Greubel, Apr 22 2017 *)

PARI

A123710 Indices n such that 4 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

4, 6, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024

Offset: 1

Paul D. Hanna, Oct 09 2006

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

Except for a(2)=6, these are proper prime powers, i.e., numbers p^k where k>1, p prime (A025475). - M. F. Hasler, Feb 12 2012

Cf. A123706, A123709, A123711, A123712; A010766.

for(n=1,1e4, A123709(n)==4 & print1(n","))  \\ M. F. Hasler, Feb 12 2012

a(n) = A025475(n) for n>2 (conjectured). - M. F. Hasler, Feb 12 2012

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A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

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A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

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A178212 Nonsquarefree numbers divisible by exactly three distinct primes.

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A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706.

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A123710 Indices n such that 4 = A123709(n) = number of nonzero terms in row n of triangle A123706.

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