A200511 -id:A200511 - cp's OEIS frontend

A178212 Nonsquarefree numbers divisible by exactly three 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, 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

A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.

Original entry on oeis.org

420, 630, 660, 780, 840, 924, 990, 1020, 1050, 1092, 1140, 1170, 1260, 1320, 1380, 1386, 1428, 1470, 1530, 1540, 1560, 1596, 1638, 1650, 1680, 1710, 1716, 1740, 1820, 1848, 1860, 1890, 1932, 1950, 1980, 2040, 2070, 2100, 2142, 2184, 2220, 2244

Offset: 1

M. F. Hasler, Feb 09 2012

nonn

I expect that A123709(a(k))=32.

M. F. Hasler, Table of n, a(n) for n = 1..18347

Cf. A200511, A178212.

Select[Range[2500], PrimeNu[#] == 4 && PrimeOmega[#] > 4 &](* Jean-François Alcover, Jun 30 2013 *)

is_A200521(n,c=4)={ omega(n)==c & bigomega(n)>c }

A303661 Powers of squarefree semiprimes that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10648, 11236, 12321, 13225, 13924, 14161, 14884

Offset: 1

Ilya Gutkovskiy, Apr 28 2018

nonn

			1089 is in the sequence because 1089 = 3^2*11^2.
1296 is in the sequence because 1296 = 2^4*3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A006881, A013929, A072774, A072777, A085155, A123711, A200511, A246547, A303606.

Select[Range[15000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && PrimeNu[#] == 2 && ! SquareFreeQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] == 2 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10000] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A303661(n):
    def g(x): return int(-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))
    def f(x): return n-1+x-sum(g(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A006881(n)-1)*A006881(n)) = Sum_{k>=2} (P(k)^2 - P(2*k))/2 = 0.07160601536406295068..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

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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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A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.

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A303661 Powers of squarefree semiprimes that are not squarefree.

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