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 1-5 of 5 results.

A147571 Numbers with exactly 4 distinct prime divisors {2,3,5,7}.

Original entry on oeis.org

210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2520, 2940, 3150, 3360, 3780, 4200, 4410, 5040, 5250, 5670, 5880, 6300, 6720, 7350, 7560, 8400, 8820, 9450, 10080, 10290, 10500, 11340, 11760, 12600, 13230, 13440, 14700, 15120, 15750, 16800
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

Links

Crossrefs

Cf. A002473, A086780, A143207, A147572, A147573, A147574, A147575, A147576, A147577, A147578, A147579, A147580.

Programs

  • Magma
    [n: n in [1..2*10^4] | PrimeDivisors(n) eq [2,3,5,7]]; // Vincenzo Librandi, Sep 15 2015
  • Mathematica
    a = {}; Do[If[EulerPhi[x]/x == 8/35, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[20000],PrimeNu[#]==4&&Max[FactorInteger[#][[;;,1]]]<11&] (* Harvey P. Dale, Nov 05 2024 *)
  • PARI
    is(n)=n%210==0 && n==2^valuation(n,2) * 3^valuation(n,3) * 5^valuation(n,5) * 7^valuation(n,7) \\ Charles R Greathouse IV, Jun 19 2016

Formula

a(n) = 210 * A002473(n). - David A. Corneth, May 14 2019
Sum_{n>=1} 1/a(n) = 1/48. - Amiram Eldar, Nov 12 2020

A147575 Numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19}.

Original entry on oeis.org

9699690, 19399380, 29099070, 38798760, 48498450, 58198140, 67897830, 77597520, 87297210, 96996900, 106696590, 116396280, 126095970, 135795660, 145495350, 155195040, 164894730, 174594420, 184294110, 193993800, 203693490, 213393180, 232792560, 242492250, 252191940
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

Links

Crossrefs

Cf. A060735, A080682, A143207, A147571-A147575, A147576-A147580.

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[9699690 x] == 1658880 x, AppendTo[a, 9699690 x]], {x, 1, 100}]; a

Formula

a(n) = 9699690 * A080682(n). - Amiram Eldar, Mar 10 2020
Sum_{n>=1} 1/a(n) = 1/1658880. - Amiram Eldar, Nov 12 2020

Extensions

More terms from Amiram Eldar, Mar 10 2020

A147572 Numbers with exactly 5 distinct prime divisors {2,3,5,7,11}.

Original entry on oeis.org

2310, 4620, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 32340, 34650, 36960, 41580, 46200, 48510, 50820, 55440, 57750, 62370, 64680, 69300, 73920, 76230, 80850, 83160, 92400, 97020, 101640, 103950, 110880, 113190, 115500, 124740, 127050
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

Links

Crossrefs

Cf. A051038, A060735, A143207, A147571-A147575, A147576-A147580.

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[x]/x == 16/77, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[130000],FactorInteger[#][[All,1]]=={2,3,5,7,11}&] (* Harvey P. Dale, Oct 04 2020 *)
  • Python
    from sympy import integer_log, prevprime
def A147572(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 g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,11)
    return 2310*bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Formula

a(n) = 2310 * A051038(n). - Amiram Eldar, Mar 10 2020
Sum_{n>=1} 1/a(n) = 1/480. - Amiram Eldar, Nov 12 2020

Extensions

More terms from Amiram Eldar, Mar 10 2020

A147574 Numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17}.

Original entry on oeis.org

510510, 1021020, 1531530, 2042040, 2552550, 3063060, 3573570, 4084080, 4594590, 5105100, 5615610, 6126120, 6636630, 7147140, 7657650, 8168160, 8678670, 9189180, 10210200, 10720710, 11231220, 12252240, 12762750, 13273260, 13783770
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

Links

Crossrefs

Cf. A060735, A080681, A143207, A147571-A147575, A147576-A147580.

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[x 510510] == 92160 x, AppendTo[a, 510510 x]], {x, 1, 100}]; a
sdpdQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Length[f]==7&&Max[f]==17]; Select[Range[510510,138*10^5,510510],sdpdQ] (* Harvey P. Dale, Aug 03 2019 *)

Formula

a(n) = 510510 * A080681(n). - Amiram Eldar, Mar 10 2020
Sum_{n>=1} 1/a(n) = 1/92160. - Amiram Eldar, Nov 12 2020

A214195 Numbers with the number of distinct prime factors a multiple of 3.

Original entry on oeis.org

1, 30, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285
Offset: 1

Views

Author

Enrique Pérez Herrero, Jul 07 2012

Keywords

Comments

If GCD(a(n),a(m))=1, then a(n)*a(m) is also in this sequence. - Enrique Pérez Herrero, Nov 23 2013

Links

Crossrefs

Subsequences include A033992, A067885, A007304 and A147573.
Cf. A145784, A030230, A030231.

Programs

  • Mathematica
    Select[Range[1000],Mod[PrimeNu[#],3]==0&]
  • PARI
    is(n)=omega(n)%3==0 \\ Charles R Greathouse IV, Sep 14 2015

Formula

A010872(A001221(a(n))) = 0.
Showing 1-5 of 5 results.

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

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