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 1-6 of 6 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

A147576 Numbers with exactly 3 distinct odd prime divisors {3,5,7}.

Original entry on oeis.org

105, 315, 525, 735, 945, 1575, 2205, 2625, 2835, 3675, 4725, 5145, 6615, 7875, 8505, 11025, 13125, 14175, 15435, 18375, 19845, 23625, 25515, 25725, 33075, 36015, 39375, 42525, 46305, 55125, 59535, 65625, 70875, 76545, 77175, 91875, 99225
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

Links

Crossrefs

Cf. A060735, A108347, A143207, A147571-A147575, A147576-A147580.

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[x]/x == 16/35, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[100000],EulerPhi[#]/#==16/35&] (* Harvey P. Dale, Dec 01 2013 *)

Formula

a(n) = 105 * A108347(n). - Amiram Eldar, Mar 10 2020
Sum_{n>=1} 1/a(n) = 1/48. - Amiram Eldar, Dec 22 2020

A147580 Numbers with exactly 7 distinct odd prime divisors {3,5,7,11,13,17,19}.

Original entry on oeis.org

4849845, 14549535, 24249225, 33948915, 43648605, 53348295, 63047985, 72747675, 82447365, 92147055, 101846745, 121246125, 130945815, 160044885, 169744575, 189143955, 218243025, 237642405, 247342095, 266741475, 276441165, 305540235, 315239925, 363738375, 373438065
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

Links

Crossrefs

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

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[4849845 x] == 1658880 x, AppendTo[a, 4849845 x]], {x, 1, 100}]; a

Formula

Sum_{n>=1} 1/a(n) = 1/1658880. - Amiram Eldar, Dec 22 2020

Extensions

More terms from Amiram Eldar, Mar 11 2020

A147577 Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.

Original entry on oeis.org

1155, 3465, 5775, 8085, 10395, 12705, 17325, 24255, 28875, 31185, 38115, 40425, 51975, 56595, 63525, 72765, 86625, 88935, 93555, 114345, 121275, 139755, 144375, 155925, 169785, 190575, 202125, 218295, 259875, 266805, 280665, 282975, 317625
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

Links

Crossrefs

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

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[x]/x == 32/77, AppendTo[a, x]], {x, 1, 1000000}]; a
Select[Range[350000],EulerPhi[#]/#==32/77&] (* Harvey P. Dale, Mar 25 2016 *)
  • Python
    from sympy import integer_log
def A147577(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):
        c = n+x
        for i11 in range(integer_log(x,11)[0]+1):
            for i7 in range(integer_log(x11:=x//11**i11,7)[0]+1):
                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 1155*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

Sum_{n>=1} 1/a(n) = 1/480. - Amiram Eldar, Dec 22 2020

A147578 Numbers with exactly 5 distinct odd prime divisors {3,5,7,11,13}.

Original entry on oeis.org

15015, 45045, 75075, 105105, 135135, 165165, 195195, 225225, 315315, 375375, 405405, 495495, 525525, 585585, 675675, 735735, 825825, 945945, 975975, 1126125, 1156155, 1216215, 1366365, 1486485, 1576575, 1756755, 1816815, 1876875, 2027025, 2147145, 2207205, 2477475
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244;
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849;
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576;
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577;
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578;
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579;
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580;
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581.

Links

Crossrefs

Cf. A000010 (EulerPhi), A060735, A143207, A147571-A147575, A147576-A147580.

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[x]/x == 384/1001, AppendTo[a, x]], {x, 1, 1000000}]; a

Formula

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Dec 22 2020

Extensions

More terms from Amiram Eldar, Mar 11 2020

A147581 Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.

Original entry on oeis.org

111546435, 334639305, 557732175, 780825045, 1003917915, 1227010785, 1450103655, 1673196525, 1896289395, 2119382265, 2342475135, 2565568005, 2788660875, 3011753745, 3681032355, 3904125225, 4350310965, 5019589575, 5465775315, 5688868185, 6135053925, 6358146795
Offset: 1

Views

Author

Artur Jasinski, Nov 07 2008

Keywords

Comments

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

Links

Crossrefs

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

Programs

  • Mathematica
    a = {}; Do[If[EulerPhi[111546435 x] == 36495360 x, AppendTo[a, 111546435 x]], {x, 1, 100}]; a
  • Python
    from sympy import integer_log
def A147581(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):
        c = n+x
        for i23 in range(integer_log(x,23)[0]+1):
            for i19 in range(integer_log(x23:=x//23**i23,19)[0]+1):
                for i17 in range(integer_log(x19:=x23//19**i19,17)[0]+1):
                    for i13 in range(integer_log(x17:=x19//17**i17,13)[0]+1):
                        for i11 in range(integer_log(x13:=x17//13**i13,11)[0]+1):
                            for i7 in range(integer_log(x11:=x13//11**i11,7)[0]+1):
                                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 111546435*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

Sum_{n>=1} 1/a(n) = 1/36495360. - Amiram Eldar, Dec 22 2020

Extensions

More terms from Amiram Eldar, Mar 11 2020
Showing 1-6 of 6 results.

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

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