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.

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A033851 Numbers whose prime factors are 5 and 7.

Original entry on oeis.org

35, 175, 245, 875, 1225, 1715, 4375, 6125, 8575, 12005, 21875, 30625, 42875, 60025, 84035, 109375, 153125, 214375, 300125, 420175, 546875, 588245, 765625, 1071875, 1500625, 2100875, 2734375, 2941225, 3828125, 4117715, 5359375, 7503125
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

Numbers k such that phi(k)/k == 24/35. - Artur Jasinski, Nov 09 2008
Subsequence of A143202. - Reinhard Zumkeller, Sep 13 2011

Links

Crossrefs

Cf. A003595, A147571-A147575, A147576-A147580. [Artur Jasinski, Nov 09 2008]
Cf. A033845, A033846, A033847, A033848, A033849, A033850, A143201, A143202.
Subsequence of A256617.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a033851 n = a033851_list !! (n-1)
a033851_list = f (singleton (5*7)) where
   f s = m : f (insert (5*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011
  • Mathematica
    a = {}; Do[If[EulerPhi[x]/x == 24/35, AppendTo[a, x]], {x, 1, 10000}]; a (* Artur Jasinski, Nov 09 2008 *)
Take[With[{nn=10},Sort[Flatten[Table[5^i 7^j,{i,nn},{j,nn}]]]],40] (* Harvey P. Dale, Feb 09 2013 *)

Formula

a(n) = 35*A003595(n). - Artur Jasinski, Nov 09 2008
A143201(a(n)) = 3. - Reinhard Zumkeller, Sep 13 2011
Sum_{n>=1} 1/a(n) = 1/24. - Amiram Eldar, Dec 22 2020

Extensions

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.

Original entry on oeis.org

15, 35, 45, 75, 135, 143, 175, 225, 245, 323, 375, 405, 675, 875, 899, 1125, 1215, 1225, 1573, 1715, 1763, 1859, 1875, 2025, 3375, 3599, 3645, 4375, 5183, 5491, 5625, 6075, 6125, 6137, 8575, 9375, 10125, 10403, 10935, 11663, 12005, 16875, 17303, 18225
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 12 2008

Keywords

Comments

Subsequence of A007774.
A037074 is a subsequence.

Examples

			a(1) = 15 = 3 * 5 = A001359(1) * A006512(1).
a(2) = 35 = 5 * 7 = A001359(2) * A006512(2).
a(3) = 45 = 3^2 * 5 = A001359(1)^2 * A006512(1).
a(4) = 75 = 3 * 5^2 = A001359(1) * A006512(1)^2.
a(5) = 135 = 3^3 * 5 = A001359(1)^3 * A006512(1).
a(6) = 143 = 11 * 13 = A001359(3) * A006512(3).
a(7) = 175 = 5^2 * 7 = A001359(2)^2 * A006512(2).
a(8) = 225 = 3^2 * 5^2 = A001359(1)^2 * A006512(1)^2.
a(9) = 245 = 5 * 7^2 = A001359(2) * A006512(2)^2.
a(10) = 323 = 17 * 19 = A001359(4) * A006512(4).
a(11) = 375 = 3 * 5^3 = A001359(1) * A006512(1)^3.
a(12) = 405 = 3^4 * 5 = A001359(1)^4 * A006512(1).

Links

Crossrefs

Cf. A001221, A001359, A006512, A006530, A007774, A020639, A037074, A143201.

Programs

  • Haskell
    a143202 n = a143202_list !! (n-1)
a143202_list = filter (\x -> a006530 x - a020639 x == 2) [1,3..]
-- Reinhard Zumkeller, Sep 13 2011
  • Mathematica
    tdpfQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==2]; Select[Range[20000],tdpfQ] (* Harvey P. Dale, Mar 04 2023 *)

Formula

A143201(a(n)) = 3.
A020639(a(n)) in A001359 and A006530(a(n)) in A006512.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/(A001359(n)^2-1) = 0.1812568234997... . - Amiram Eldar, Oct 26 2024

A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.

Original entry on oeis.org

21, 63, 77, 147, 189, 221, 437, 441, 539, 567, 847, 1029, 1323, 1517, 1701, 2021, 2873, 3087, 3757, 3773, 3969, 4757, 5103, 5929, 6557, 7203, 8303, 9261, 9317, 9797, 10051, 11021, 11907, 12317, 15309, 16637, 21609
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 12 2008

Keywords

Comments

Subsequence of A007774.
A033850 is a subsequence.
Subsequence of A195106. - Reinhard Zumkeller, Sep 13 2011

Examples

			a(1) = 21 = 3 * 7 = A023200(1) * A046132(1).
a(2) = 63 = 3^2 * 7 = A023200(1)^2 * A046132(1).
a(3) = 77 = 7 * 11 = A023200(2) * A046132(2).
a(4) = 147 = 3 * 7^2 = A023200(1) * A046132(1)^2.
a(5) = 189 = 3*3 * 7 = A023200(1)^3 * A046132(1).
a(6) = 221 = 13 * 17 = A023200(3) * A046132(3).
a(7) = 437 = 19 * 23 = A023200(4) * A046132(4).
a(8) = 441 = 3^2 * 7^2 = A023200(1)^2 * A046132(1)^2.
a(9) = 539 = 7^2 * 11 = A023200(2)^2 * A046132(2).
a(10) = 567 = 3^4 * 7 = A023200(1)^4 * A046132(1).

Links

Crossrefs

Cf. A001221, A006530, A007774, A020639, A023200, A027748, A033850, A046132, A143201, A195106.

Programs

  • Haskell
    a143203 n = a143203_list !! (n-1)
a143203_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 4 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011
  • Mathematica
    dpf2Q[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==4]; Select[Range[22000],dpf2Q] (* Harvey P. Dale, Mar 18 2023 *)

Formula

A143201(a(n)) = 5.
A020639(a(n)) in A023200 and A006530(a(n)) in A046132.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023200(n)+1)^2-4) = 0.109882433872... . - Amiram Eldar, Oct 26 2024

A143204 Union of A143207 and A033847.

Original entry on oeis.org

14, 28, 30, 56, 60, 90, 98, 112, 120, 150, 180, 196, 224, 240, 270, 300, 360, 392, 448, 450, 480, 540, 600, 686, 720, 750, 784, 810, 896, 900, 960, 1080, 1200, 1350, 1372, 1440, 1500, 1568, 1620, 1792, 1800, 1920, 2160, 2250, 2400, 2430, 2700, 2744, 2880
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 12 2008

Keywords

Comments

Subsequence of A195238. - Harvey P. Dale, Sep 13 2011

Examples

			a(1)  =  14 = 2 * 7 = A033847(1).
a(2)  =  28 = 2^2 * 7 = A033847(2).
a(3)  =  30 = 2 * 3 * 5 = A143207(1).
a(4)  =  56 = 2^3 * 7 = A033847(3).
a(5)  =  60 = 2^2 * 3 * 5 = A143207(2).
a(6)  =  90 = 2 * 3^2 * 5 = A143207(3).
a(7)  =  98 = 2 * 7^2 = A033847(4).
a(8)  = 112 = 2^4 * 7 = A033847(5).
a(9)  = 120 = 2^3 * 3 * 5 = A143207(4).
a(10) = 150 = 2 * 3 * 5^2 = A143207(5).
a(11) = 180 = 2^2 * 3^2 * 5 = A143207(6).
a(12) = 196 = 2^2 * 7^2 = A033847(6).

Links

Crossrefs

Cf. A033847, A143201, A143207, A195238.

Programs

  • Mathematica
    q[n_] := Module[{p1 = {2, 3, 5}, p2 = {2, 7}, e1, e2}, e1 = IntegerExponent[n, p1]; e2 = IntegerExponent[n, p2]; (Times @@ e1 > 0 && Times @@ (p1^e1) == n) || (Times @@ e2 > 0 && Times @@ (p2^e2) == n)]; Select[Range[3000], q] (* Amiram Eldar, Oct 25 2024 *)

Formula

A143201(a(n)) = 6. - Harvey P. Dale, Sep 13 2011
Sum_{n>=1} 1/a(n) = 7/24. - Amiram Eldar, Oct 25 2024

Extensions

Corrected by Harvey P. Dale, Aug 21 2011
Revised version with improved definition; thanks to Harvey P. Dale, who noticed that the original definition was not sufficient.

A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

Original entry on oeis.org

55, 91, 187, 247, 275, 391, 605, 637, 667, 1147, 1183, 1375, 1591, 1927, 2057, 2491, 3025, 3127, 3179, 3211, 4087, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7387, 8281, 8993, 9991, 10807, 11227, 12091, 15125, 15341, 15379, 17947, 19343, 22627, 23707
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 30 2008

Keywords

Comments

Subsequence of A007774.
A111192 is a subsequence.
Subsequence of A195118. - Reinhard Zumkeller, Sep 13 2011

Examples

			a(1) = 55 = 5 * 11 = A023201(1) * A046117(1).
a(2) = 91 = 7 * 13 = A023201(2) * A046117(2).
a(3) = 187 = 11 * 17 = A023201(3) * A046117(3).
a(4) = 247 = 13 * 19 = A023201(4) * A046117(4).
a(5) = 275 = 5^2 * 11 = A023201(1)^2 * A046117(1).
a(6) = 391 = 17 * 23 = A023201(5) * A046117(5).
a(7) = 605 = 5 * 11^2 = A023201(1) * A046117(1)^2.
a(8) = 637 = 7^2 * 13 = A023201(2)^2 * A046117(2).
a(9) = 667 = 23 * 29 = A023201(6) * A046117(6).
a(10) = 1147 = 31 * 37 = A023201(7) * A046117(7).

Links

Crossrefs

Cf. A001221, A006530, A007774, A020639, A023201, A027748, A046117, A111192, A143201, A195118.

Programs

  • Haskell
    a143205 n = a143205_list !! (n-1)
a143205_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 6 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011
  • Mathematica
    okQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]},Length[fi]==2 && Last[fi]-First[fi]==6]; Select[Range[25000],okQ]  (* Harvey P. Dale, Apr 18 2011 *)

Formula

A143201(a(n)) = 7.
A020639(a(n)) in A023201 and A006530(a(n)) in A046117.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023201(n)+2)^2-9) = 0.058842810164... . - Amiram Eldar, Oct 26 2024
Previous Showing 11-15 of 15 results.

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