A108347 -id:A108347

Search: a108347 -id:a108347

Displaying 1-8 of 8 results found.

page 1

A003593

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

+10
45

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

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

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

MATHEMATICA

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

PROG

(PARI) 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

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

(Haskell)

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

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

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

(Python)

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

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

A147576

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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

FORMULA

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

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

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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Artur Jasinski, Nov 07 2008

STATUS

approved

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].

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=0..15.

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.

EXAMPLE

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

CROSSREFS

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

KEYWORD

fini,nonn

AUTHOR

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

STATUS

approved

A108513

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

A342950

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

(Python)

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

(Python)

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

(Python) # 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

CROSSREFS

Union of A108319 and A108347.

Intersection of A002473 and A067251.

KEYWORD

nonn

AUTHOR

David A. Corneth, Mar 30 2021

STATUS

approved

A194360

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

EXAMPLE

The triangle has rows beginning with 3^k and ending with 105^k:

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

MATHEMATICA

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

CROSSREFS

Cf. A194356, A194357, A194358, A194359.

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

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, Sep 08 2011

STATUS

approved

A362012

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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

CROSSREFS

Cf. A007947, A108347, A236206.

KEYWORD

nonn

AUTHOR

Michael De Vlieger, Apr 04 2023

STATUS

approved

A241681

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

+10
0

2, 3, 5, 7, 735, 2333772

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..6.

EXAMPLE

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

MAPLE

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:

CROSSREFS

Cf. A108319, A108347.

Subsequence of A046034.

KEYWORD

nonn,base,hard

AUTHOR

Michel Lagneau, Apr 27 2014

STATUS

approved

page 1

Search completed in 0.013 seconds