A108347 -id:A108347 - cp's OEIS frontend

A003593 Numbers of the form 3^i*5^j with i, j >= 0.

1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243, 375, 405, 625, 675, 729, 1125, 1215, 1875, 2025, 2187, 3125, 3375, 3645, 5625, 6075, 6561, 9375, 10125, 10935, 15625, 16875, 18225, 19683, 28125, 30375, 32805, 46875, 50625, 54675, 59049

Offset: 1

N. J. A. Sloane

nonn

Odd 5-smooth numbers (A051037). - Reinhard Zumkeller, Sep 18 2005

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

Cf. A033849, A112751-A112756, A143202, A022337 (list of j), A022336(list of i).

Cf. A264997 (partitions into), see also A264998. Cf. A108347 (odd 7-smooth).

Filtered([1..60000],n->PowerMod(15,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003593 n = a003593_list !! (n-1)
a003593_list = f (singleton 1) where
   f s = m : f (insert (3*m) $ insert (5*m) s') where
     (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 13 2011

[n: n in [1..60000] | PrimeDivisors(n) subset [3,5]]; // Bruno Berselli, Sep 24 2012

isA003593 := proc(n)
    if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {3, 5} = {} );
    end if;
end proc:
A003593 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA003593(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003593(n),n=1..30) ; # R. J. Mathar, Aug 04 2016

fQ[n_] := PowerMod[15, n, n] == 0; Select[Range[60000], fQ] (* Bruno Berselli, Sep 24 2012 *)

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(5),N=5^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

is(n)=n==3^valuation(n,3)*5^valuation(n,5) \\ Charles R Greathouse IV, Apr 23 2013

from sympy import integer_log
def A003593(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 n+x-sum(integer_log(x//5**i,3)[0]+1 for i in range(integer_log(x,5)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

a(n) ~ 1/sqrt(15)*exp(sqrt(2*log(3)*log(5)*n)) asymptotically. - Benoit Cloitre, Jan 22 2002
The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(15*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
Sum_{n>=1} 1/a(n) = (3*5)/((3-1)*(5-1)) = 15/8. - Amiram Eldar, Sep 22 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

Artur Jasinski, Nov 07 2008

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

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

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 *)

a(n) = 105 * A108347(n). - Amiram Eldar, Mar 10 2020

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

A146205 Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n odd (n=15 in this example), is equal to half-integer values k+1/2, -[n/2]-1<=k<=[n/2].

Original entry on oeis.org

35, 35, 245, 245, 735, 735, 1225, 1225, 1225, 1225, 735, 735, 245, 245, 35, 35

Offset: 0

Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Oct 28 2008, May 04 2010

1) Closed-form expressions for sequences see Pfeifer (2010).
2) The median taken on partial sums of the simple random walk represents the market price in a simulation model wherein a single security among non-cooperating and asymetrically informed traders is traded (Pfeifer et al. 2009).
3) A146207=A146205+(0,A146206) see lemma 2 in Pfeifer (2010).

			All possible different paths (sequences of partial sums) in case of n=3:
{0,-1,-2,-3}; median=-1.5
{0,-1,-2,-1}; median=-1
{0,-1,0,-1}; median=-0.5
{0,-1,0,1}; median=0
{0,1,0,-1}; median=0
{0,1,0,1}; median=0.5
{0,1,2,1}; median=1
{0,1,2,3}; median=1.5
sequence of integers in case of n=3: 1,1,1,1

Pfeifer, C. (2010) Probability distribution of the median taken on partial sums of the simple random walk, Submitted to Stochastic Analysis and Applications

C. Pfeifer, K. Schredelseker, G. U. H. Seeber, On the negative value of information in informationally inefficient markets. Calculations for large number of traders, Eur. J. Operat. Res., 195 (1) (2009) 117-126.

A117692, A108347, A029460, A029466, A135553, A137272, A146206, A146207

A108513 Numbers of the form (2^i)(5^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 25, 28, 32, 35, 40, 49, 50, 56, 64, 70, 80, 98, 100, 112, 125, 128, 140, 160, 175, 196, 200, 224, 245, 250, 256, 280, 320, 343, 350, 392, 400, 448, 490, 500, 512, 560, 625, 640, 686, 700, 784, 800, 875, 896, 980, 1000, 1024, 1120

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 05 2005

Numbers m | 70^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051037, A108319, A108347, A003592, A003591, A003595.

With[{n = 1120}, Sort@ Flatten@ Table[2^i * 5^j * 7^k, {i, 0, Log2@ n}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(2^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

isok(n) = (n/(2^valuation(n,2)*5^valuation(n,5)*7^valuation(n,7)) == 1); \\ Michel Marcus, Oct 01 2013

Sum_{n>=1} 1/a(n) = (2*5*7)/((2-1)*(5-1)*(7-1)) = 35/12. - Amiram Eldar, Sep 23 2020

a(n) ~ exp((6*log(2)*log(5)*log(7)*n)^(1/3)) / sqrt(70). - Vaclav Kotesovec, Sep 23 2020

A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 105, 108, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324

Offset: 1

David A. Corneth, Mar 30 2021

nonn

			12 is in the sequence as all of its prime divisors are <= 7 and 12 is not divisible by 10.

David A. Corneth, Table of n, a(n) for n = 1..10195

Union of A108319 and A108347.

Intersection of A002473 and A067251.

Select[Range@500,Max[First/@FactorInteger@#]<=7&&Mod[#,10]!=0&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

is(n) = if(n%10 == 0, return(0)); forprime(p = 2, 7, n/=p^valuation(n, p)); n==1

A342950_list, n = [], 1
while n < 10**9:
    if n % 10:
        m = n
        for p in (2,3,5,7):
            q, r = divmod(m,p)
            while r == 0:
                m = q
                q, r = divmod(m,p)
        if m == 1:
            A342950_list.append(n)
    n += 1 # Chai Wah Wu, Mar 31 2021

from sympy import integer_log
def A342950(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,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,3)[0]+1):
                c -= (k:=m//3**j).bit_length()+integer_log(k,5)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 17 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A342950gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                if not (p==2 and v%5==0) and not (p==5 and v&1==0):
                    heapq.heappush(h, v*p)
print(list(islice(A342950gen(), 65))) # Michael S. Branicky, Sep 17 2024

Sum_{n>=1} 1/a(n) = 63/16. - Amiram Eldar, Apr 01 2021

A194360 Triangle of divisors of 105^n, each number occurring once.

Original entry on oeis.org

1, 3, 5, 7, 15, 21, 35, 105, 9, 25, 45, 49, 63, 75, 147, 175, 225, 245, 315, 441, 525, 735, 1225, 1575, 2205, 3675, 11025, 27, 125, 135, 189, 343, 375, 675, 875, 945, 1029, 1125, 1323, 1715, 2625, 3087, 3375, 4725, 5145, 6125, 6615, 7875, 8575, 9261, 15435

Offset: 0

T. D. Noe, Sep 08 2011

The length of row k is A003215(k), the centered hexagonal numbers, 3k^2 + 3k + 1.

			The triangle has rows beginning with 3^k and ending with 105^k:
1
3, 5, 7, 15, 21, 35, 105
9, 25, 45, 49, 63, 75, 147, 175, 225, 245, 315, 441, 525, 735, 1225, 1575, 2205, 3675, 11025

T. D. Noe, Rows n = 0..20

Cf. A194356, A194357, A194358, A194359.

Cf. A108347 (numbers of the form (3^i)*(5^j)*(7^k))

Join[{{1}}, Table[Complement[Divisors[105^n], Divisors[105^(n-1)]], {n, 9}]]

A362012 Numbers k such that 1 < gcd(k, 105) < k and A007947(k) does not divide 105.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 28, 30, 33, 36, 39, 40, 42, 48, 50, 51, 54, 55, 56, 57, 60, 65, 66, 69, 70, 72, 77, 78, 80, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 108, 110, 111, 112, 114, 115, 117, 119, 120, 123, 126, 129, 130, 132, 133, 138, 140, 141, 144, 145, 150, 153, 154, 155, 156, 159, 160

Offset: 1

Michael De Vlieger, Apr 04 2023

nonn

The asymptotic density of this sequence is 19/35. - Amiram Eldar, Dec 02 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram showing numbers k in this sequence instead as k mod 105, in black, else white if k is coprime to 105, purple if k = 1, red if k | 105, and gold if rad(k) | 105, magnification 10X.

Cf. A007947, A108347, A236206.

With[{n = 105}, Select[Range[200], And[! CoprimeQ[#, n], ! Divisible[n, Times @@ FactorInteger[#][[All, 1]]]] & ] ]

This sequence is { N \ { A108347 U A236206 } }.

A241681 Numbers n such that the decimal digits of n are also the prime divisors of n.

Original entry on oeis.org

2, 3, 5, 7, 735, 2333772

Offset: 1

Michel Lagneau, Apr 27 2014

The sequence is given for a(n) < 10^11.
No more terms <= 10^150. Terms are of the form 2^e2 * 3^e3 * 7^e7 or of the form 3^e3 * 5^e5 * 7^e7, for which no other number <= 10^150 than those listed is a term. - David A. Corneth, Sep 28 2019
No more terms <= 10^1000. - Michael S. Branicky, May 30 2025

			735 = 3*5*7^2 is in the sequence because the digits 7, 3 and 5 are also the prime divisors of 735.

Cf. A108319, A108347.

Subsequence of A046034.

with(numtheory):nn:=1000000:for n from 1 to 10^11 do:lst:={}:x:=factorset(n):y:=convert(n,base,10):n1:=nops(x):n2:=nops(y): for j from 1 to n2 do:lst:=lst union {y[j]}:od:if x=lst then print(n):else fi:od:

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A003593 Numbers of the form 3^i*5^j with i, j >= 0.

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

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A146205 Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n odd (n=15 in this example), is equal to half-integer values k+1/2, -[n/2]-1<=k<=[n/2].

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A108513 Numbers of the form (2^i)*(5^j)*(7^k), with i, j, k >= 0.

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A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.

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A194360 Triangle of divisors of 105^n, each number occurring once.

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A362012 Numbers k such that 1 < gcd(k, 105) < k and A007947(k) does not divide 105.

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A108513 Numbers of the form (2^i)(5^j)(7^k), with i, j, k >= 0.