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.

A046389 Products of exactly three distinct odd primes.

Original entry on oeis.org

105, 165, 195, 231, 255, 273, 285, 345, 357, 385, 399, 429, 435, 455, 465, 483, 555, 561, 595, 609, 615, 627, 645, 651, 663, 665, 705, 715, 741, 759, 777, 795, 805, 861, 885, 897, 903, 915, 935, 957, 969, 987, 1001, 1005, 1015, 1023, 1045, 1065, 1085
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Comments

The old name was "Odd numbers with exactly 3 distinct prime factors", which is slightly ambiguous, since it might be interpreted to include 315 = 3^2*5*7. Cf. A278569. - N. J. A. Sloane, Nov 27 2016

Links

Crossrefs

Cf. A046316, A046405, A007304, A278569.

Programs

  • Mathematica
    f[n_] := OddQ[n] && Last/@FactorInteger[n]=={1,1,1}; lst={}; Do[If[f[n], AppendTo[lst,n]], {n, 2000}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
  • PARI
    list(lim)=my(v=List(), t); forprime(p=3, lim^(1/3), forprime(q=p+1, sqrt(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 26 2011
  • Python
    from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A046389(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Extensions

Name clarified by N. J. A. Sloane, Nov 27 2016

A046316 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

Original entry on oeis.org

27, 45, 63, 75, 99, 105, 117, 125, 147, 153, 165, 171, 175, 195, 207, 231, 245, 255, 261, 273, 275, 279, 285, 325, 333, 343, 345, 357, 363, 369, 385, 387, 399, 423, 425, 429, 435, 455, 465, 475, 477, 483, 507, 531, 539, 549, 555, 561, 575, 595, 603, 605
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Links

Crossrefs

A369979 sorted into ascending order.
Subsequence of A014612 and of A046340.
Cf. A255646 (final digits), A369054, A369058 (characteristic function), A369252 [= A003415(a(n))].
Subsequences: A046389, A046373, A046405, A075814, A338469, A338556, A338557, A369246.

Programs

  • Haskell
    a046316 n = a046316_list !! (n-1)
a046316_list = filter ((== 3) . a001222) [1, 3 ..]
-- Reinhard Zumkeller, May 05 2015
  • PARI
    list(lim)=my(v=List(),pq); forprime(p=3,lim\9, forprime(q=3,min(lim\3\p,p), pq=p*q; forprime(r=3,lim\pq, listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A046316(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b+1 for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

Extensions

Definition clarified by N. J. A. Sloane, Dec 19 2017

A046373 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd palindromic primes (odd terms from A002385).

Original entry on oeis.org

27, 45, 63, 75, 99, 105, 125, 147, 165, 175, 231, 245, 275, 343, 363, 385, 539, 605, 847, 909, 1179, 1331, 1359, 1515, 1629, 1719, 1965, 2121, 2265, 2525, 2715, 2751, 2817, 2865, 3171, 3177, 3275, 3333, 3357, 3447, 3535, 3775, 3801, 4011, 4323, 4525
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Crossrefs

Cf. A002385, A046316, A046405.

Programs

  • Mathematica
    Take[Times@@@Tuples[Select[Prime[Range[2,100]],PalindromeQ],3]//Union,50] (* Harvey P. Dale, Sep 10 2019 *)

Extensions

Definition clarified by N. J. A. Sloane, Dec 19 2017 at the suggestion of Harvey P. Dale
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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