A143204 -id:A143204 - cp's OEIS frontend

A143207 Numbers with distinct prime factors 2, 3, and 5.

30, 60, 90, 120, 150, 180, 240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810, 900, 960, 1080, 1200, 1350, 1440, 1500, 1620, 1800, 1920, 2160, 2250, 2400, 2430, 2700, 2880, 3000, 3240, 3600, 3750, 3840, 4050, 4320, 4500, 4800, 4860

Offset: 1

Reinhard Zumkeller, Aug 12 2008

Numbers of the form 2^i * 3^j * 5^k with i, j, k > 0. - Reinhard Zumkeller, Sep 13 2011

Integers k such that phi(k)/k = 4/15. - Artur Jasinski, Nov 07 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A069819.

Subsequence of A143204 and of A051037.

import Data.Set (singleton, deleteFindMin, insert)
a143207 n = a143207_list !! (n-1)
a143207_list = f (singleton (2*3*5)) where
   f s = m : f (insert (2*m) $ insert (3*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

[n: n in [1..5000] | PrimeDivisors(n) eq [2,3,5]]; // Bruno Berselli, Sep 14 2015

a = {}; Do[If[EulerPhi[x]/x == 4/15, AppendTo[a, x]], {x, 1, 11664}]; a (* Artur Jasinski, Nov 07 2008 *)
n = 10^4; Table[2^i*3^j*5^k, {i, 1, Log[2, n]}, {j, 1, Log[3, n/2^i]}, {k, 1, Log[5, n/(2^i*3^j)]}] // Flatten // Sort (* Amiram Eldar, Sep 24 2020 *)

list(lim)=my(v=List(),s,t); for(i=1,logint(lim\6,5), t=5^i; for(j=1,logint(lim\t\2,3), s=t*3^j; while((s<<=1)<=lim, listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

is(n) = if(n%30,return(0)); my(f=factor(n,6)[,1]); f[#f]<6 \\ David A. Corneth, Sep 22 2020

from sympy import integer_log
def A143207(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 i in range(integer_log(x,5)[0]+1):
            for j in range(integer_log(m:=x//5**i,3)[0]+1):
                c -= (m//3**j).bit_length()
        return c
    return bisection(f,n,n)*30 # Chai Wah Wu, Sep 16 2024

A001221(a(n)) = 3; A020639(a(n)) = 2; A006530(a(n)) = 5; A143201(a(n)) = 6.
a(n) = 30*A051037(n); A007947(a(n)) = A010869(n). - Reinhard Zumkeller, Sep 13 2011
a(n) ~ sqrt(30) * exp((6*log(2)*log(3)*log(5)*n)^(1/3)). - Vaclav Kotesovec, Sep 22 2020
Sum_{n>=1} 1/a(n) = 1/8. - Amiram Eldar, Sep 24 2020

New name from Charles R Greathouse IV, Sep 14 2015

A033847 Numbers whose prime factors are 2 and 7.

Original entry on oeis.org

14, 28, 56, 98, 112, 196, 224, 392, 448, 686, 784, 896, 1372, 1568, 1792, 2744, 3136, 3584, 4802, 5488, 6272, 7168, 9604, 10976, 12544, 14336, 19208, 21952, 25088, 28672, 33614, 38416, 43904, 50176, 57344, 67228, 76832, 87808, 100352, 114688

Offset: 1

Jeff Burch

nonn

Numbers k such that phi(k) = (3/7)*k - Benoit Cloitre, Apr 19 2002

Subsequence of A143204. - Reinhard Zumkeller, Sep 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A033845, A033846, A033848, A033849, A033850, A033851, A143204.

import Data.Set (singleton, deleteFindMin, insert)
a033847 n = a033847_list !! (n-1)
a033847_list = f (singleton (2*7)) where
   f s = m : f (insert (2*m) $ insert (7*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

With[{nn=20},Select[Union[Flatten[Table[2^n 7^k,{n,nn},{k,nn}]]],#<=2^nn 7&]] (* Harvey P. Dale, Nov 25 2020 *)

A143201(a(n)) = 6. - Reinhard Zumkeller, Sep 13 2011

Sum_{n>=1} 1/a(n) = 1/6. - Amiram Eldar, Dec 22 2020

Offset fixed by Reinhard Zumkeller, Sep 13 2011

A143201 Product of distances between prime factors in factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 3, 1, 1, 2, 1, 4, 5, 10, 1, 2, 1, 12, 1, 6, 1, 6, 1, 1, 9, 16, 3, 2, 1, 18, 11, 4, 1, 10, 1, 10, 3, 22, 1, 2, 1, 4, 15, 12, 1, 2, 7, 6, 17, 28, 1, 6, 1, 30, 5, 1, 9, 18, 1, 16, 21, 12, 1, 2, 1, 36, 3, 18, 5, 22, 1, 4, 1, 40, 1, 10, 13, 42, 27, 10, 1, 6, 7, 22

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

a(n) is the product of the sum of 1 and first differences of prime factors of n with multiplicity, with a(n) = 1 for n = 1 or prime n. - Michael De Vlieger, Nov 12 2023.
a(A007947(n)) = a(n);
A006093 and A001747 give record values and where they occur:
A006093(n)=a(A001747(n+1)) for n>1.
a(n) = 1 iff n is a prime power: a(A000961(n))=1;
a(n) = 2 iff n has exactly 2 and 3 as prime factors:
a(A033845(n))=2;
a(n) = 3 iff n is in A143202;
a(n) = 4 iff n has exactly 2 and 5 as prime factors:
a(A033846(n))=4;
a(n) = 5 iff n is in A143203;
a(n) = 6 iff n is in A143204;
a(n) = 7 iff n is in A143205;
a(n) <> A006512(k)+1 for k>1.
a(A033849(n))=3; a(A033851(n))=3; a(A033850(n))=5; a(A033847(n))=6; a(A033848(n))=10. [Reinhard Zumkeller, Sep 19 2011]

			a(86) = a(43*2) = 43-2+1 = 42;
a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A000961, A001747, A006093, A020639, A033845, A033846.
Cf. A033847, A033848, A033849, A033850.
Cf. A143203, A143204, A143205.
Cf. A027746.

a143201 1 = 1
a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
   where pfs = a027748_row n
-- Reinhard Zumkeller, Sep 13 2011

Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* Harvey P. Dale, Dec 07 2011 *)

a(n) = f(n,1,1) where f(n,q,y) = if n=1 then y else if q=1 then f(n/p,p,1)) else f(n/p,p,y*(p-q+1)) with p = A020639(n) = smallest prime factor of n.

A195238 Numbers with at least 2 and not more than 3 distinct prime factors not greater than 7 that are multiples of 7 or of 15.

Original entry on oeis.org

14, 15, 21, 28, 30, 35, 42, 45, 56, 60, 63, 70, 75, 84, 90, 98, 105, 112, 120, 126, 135, 140, 147, 150, 168, 175, 180, 189, 196, 224, 225, 240, 245, 252, 270, 280, 294, 300, 315, 336, 350, 360, 375, 378, 392, 405, 441, 448, 450, 480, 490, 504, 525, 540, 560

Offset: 1

Harvey P. Dale, Sep 13 2011

A143204 is a subsequence.

Subsequence of A002473.

			a(10) = 60 = 2^2 * 3 * 5.
a(11) = 63 = 3^2 * 7.
a(12) = 70 = 2 * 5 * 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001221, A006530, A143204, A002473.

a195238 n = a195238_list !! (n-1)
a195238_list = filter (\x -> a001221 x `elem` [2,3] &&
                             a006530 x `elem` [5,7] &&
                             (mod x 7 == 0 || mod x 15 == 0)) [1..]
-- Reinhard Zumkeller, Sep 13 2011

pfsQ[n_]:=Module[{fs=Transpose[FactorInteger[n]][[1]]},Max[fs]<8 && 1Harvey P. Dale, Aug 21 2011 *)

is(n)=my(v=apply(p->valuation(n,p), [2,3,5,7])); n==2^v[1]*3^v[2]*5^v[3]*7^v[4] && (v[4] || v[2]*v[3]) && factorback(v)==0 && !!v[1]+!!v[2]+!!v[3]+!!v[4]>1 \\ Charles R Greathouse IV, Sep 14 2015

2 <= A001221(a(n)) <= 3.
5 <= A006530(a(n)) <= 7.
Sum_{n>=1} 1/a(n) = 11/16. - Amiram Eldar, Oct 25 2024

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A143207 Numbers with distinct prime factors 2, 3, and 5.

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A033847 Numbers whose prime factors are 2 and 7.

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A143201 Product of distances between prime factors in factorization of n.

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A195238 Numbers with at least 2 and not more than 3 distinct prime factors not greater than 7 that are multiples of 7 or of 15.

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Haskell

Mathematica

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