cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Reinhard Zumkeller, May 24 2010

Keywords

Examples

			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.

Links

Crossrefs

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.

Programs

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

Formula

A001221(a(n)) = 3; A001222(a(n)) > 3; A000005(n) >= 12;
a(n) = A123712(n) for n <= 52, possibly more.

A200511 Numbers n with omega(n)=2 and bigomega(n)>2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity.

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, 232, 236
Offset: 1

Views

Author

M. F. Hasler, Feb 09 2012

Keywords

Comments

Equivalently, numbers of the form n=p^k*q^m where k,m>0, k+m>2 and p,q prime.
It appears that this is equal to A123711.

Links

Crossrefs

Cf. A200521, A178212.

Programs

  • Mathematica
    Select[Range[240], PrimeNu[#] == 2 && PrimeOmega[#] > 2 &] (* Jean-François Alcover, Jun 29 2013 *)
  • PARI
    for(n=1,999,bigomega(n)>2 & omega(n)==2 & print1(n","))

A380432 Numbers k such that bigomega(k) > omega(k) > 3.

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, 2280, 2340, 2380
Offset: 1

Views

Author

Michael De Vlieger, Jan 29 2025

Keywords

Comments

A200521 is a proper subset; a(144) = 4620 is not contained in A200521; A200521(144) = 4650.
Subset of A375055, which is in turn a subset of A126706.

Examples

			Table of select a(n) showing exponents listed in columns of primes written vertically in the heading. For p^0 we instead write "." for clarity:
              Exponent of prime
                     1 1 1 1
   n   a(n)   2 3 5 7 1 3 7 9
  -----------------------------
    1    420   2 1 1 1
    2    630   1 2 1 1
    3    660   2 1 1 . 1
    4    780   2 1 1 . . 1
    5    840   3 1 1 1
    6    924   2 1 . 1 1
    7    990   1 2 1 . 1
    8   1020   2 1 1 . . . 1
    9   1050   1 1 2 1
   10   1092   2 1 . 1 . 1
   11   1140   2 1 1 . . . . 1
   12   1170   1 2 1 . . 1
  144   4620   2 1 1 1 1
  190   5460   2 1 1 1 . 1
  275   6930   1 2 1 1 1

Links

Crossrefs

Cf. A001221, A001222, A126706, A200521, A375055.

Programs

  • Mathematica
    Select[Range[2500], PrimeOmega[#] > PrimeNu[#] > 3 &]
Showing 1-3 of 3 results.

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