A022336 -id:A022336 - 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

A022337 Exponent of 5 (value of j) in n-th number of form 3^i*5^j.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 3, 1, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 0, 5, 3, 1, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 0, 11, 9, 7, 5

Offset: 1

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003593, A022336, A112754, A112762, A112765.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; IntegerExponent[#, 5] & /@ Union[s] (* Amiram Eldar, Feb 06 2020 *)

a(n) = A112765(A003593(n)) = A112754(n) - A022336(n). - Reinhard Zumkeller, Sep 18 2005

A112754 Total number of prime factors of n-th number of the form 3^i*5^j.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 4, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 5, 6, 7, 6, 7, 8, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 7, 8, 9, 10, 7, 8, 9, 10, 8, 9, 10, 11, 8, 9, 10, 11, 8, 9, 10, 11, 12, 9, 10, 11, 12, 9, 10, 11, 12, 13, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 10, 11, 12, 13, 14, 10, 11, 12

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A003593, A022336, A022337, A069352, A112753, A112759.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; PrimeOmega[Union[s]] (* Amiram Eldar, Feb 06 2020 *)

a(n) = A001222(A003593(n)) = A022336(n) + A022337(n).

A112761 Exponent of 3 (value of j) in n-th number of the form 2^i3^j5^k.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 2, 0, 2, 1, 0, 3, 1, 0, 2, 1, 0, 4, 2, 1, 0, 3, 1, 0, 0, 3, 2, 1, 0, 4, 2, 1, 0, 3, 2, 1, 5, 0, 0, 3, 2, 1, 0, 4, 2, 1, 1, 0, 4, 3, 2, 1, 5, 0, 0, 3, 2, 1, 0, 0, 4, 3, 2, 6, 1, 1, 0, 4, 3, 2, 1, 5, 0, 0, 3, 2, 2, 1, 5, 0, 0, 4, 3, 2, 6, 1, 1, 0, 4, 3, 2, 1

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007949, A022336, A022329, A051037, A112759, A112760, A112762.

IntegerExponent[#, 3] & /@ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A007949(A051037(n));

a(n) = A112759(n) - A112760(n) - A112762(n).

A131579 Period 10: repeat 0, 3, 6, 9, 2, 5, 8, 1, 4, 7.

Original entry on oeis.org

0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2

Offset: 0

Paul Curtz, Oct 02 2007

a(n) = last digit of 3*n, n >= 0. a(n+7) = last digit of 3*n + 1, n >= 0, and a(n+4) = last digit of 3*n + 2, n >= 0. The last digit of -3*n = A144468(n), n >= 0, the last one of -3*n + 1 = A144468(n+3), n >= 0 and the last one of -3*n + 2 = A144468(n+6), n >= 0. - Wolfdieter Lang, Aug 14 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1)

Cf. A022336, A025648, A025655, A059626, A144468.

&cat[[0,3,6,9,2,5,8,1,4,7]^^10]; // Vincenzo Librandi, Aug 15 2017

PadRight[{}, 120, {0, 3, 6, 9, 2, 5, 8, 1, 4, 7}] (* Vincenzo Librandi, Aug 15 2017 *)

a(n) = A008585(n) mod 10. - Bruno Berselli, Mar 08 2011
G.f.: -x*(3 + 6*x + 9*x^2 + 2*x^3 + 5*x^4 + 8*x^5 + x^6 + 4*x^7 + 7*x^8) / ( (x-1) *(1+x) *(x^4 + x^3 + x^2 + x + 1) *(x^4 - x^3 + x^2 - x + 1) ). - R. J. Mathar, Mar 10 2011
Decimal expansion of 41028683/1111111110 . - R. J. Mathar, Jul 06 2025

A112753 Number of distinct prime factors of n-th number of the form 3^i*5^j.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A003593, A022336, A022337, A086412, A112754, A112758.

With[{nn=120},Take[PrimeNu/@Union[Flatten[{3^#[[1]] 5^#[[2]],5^#[[1]] 3^#[[2]]}&/@Tuples[Range[0,nn],2]]],nn]] (* Harvey P. Dale, Nov 29 2013 *)

a(n) = A001221(A003593(n)) = 2 - 0^A022336(n) + 0^A022337(n);

a(n) <= 2.

A356624 After n iterations of the "Square Multiscale" substitution, the largest tiles have side length 3^t / 5^f; a(n) = t (A356625 gives corresponding f's).

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, 10, 7, 4, 1, 11, 8, 5, 2, 12, 9, 6, 3, 0, 13, 10, 7, 4, 1, 14, 11, 8, 5, 2, 15, 12, 9, 6, 3, 0, 16, 13, 10, 7, 4, 1, 17, 14, 11, 8, 5, 2, 18, 15, 12, 9, 6, 3, 0, 19, 16, 13, 10, 7, 4, 1, 20, 17

Offset: 0

Rémy Sigrist, Aug 17 2022

nonn

See A329919 for further details about the "Square Multiscale" substitution.

			The first terms, alongside the corresponding side lengths, are:
  n   a(n)  Side length
  --  ----  -----------
   0     0            1
   1     1          3/5
   2     2         9/25
   3     3       27/125
   4     0          1/5
   5     4       81/625
   6     1         3/25
   7     5     243/3125
   8     2        9/125
   9     6    729/15625
  10     3       27/625

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Yotam Smilansky and Yaar Solomon, Multiscale Substitution Tilings, arXiv:2003.11735 [math.DS], 2020.

Cf. A022336, A329919, A354535, A356625.

{ sc = [1]; for (n=0, 78, s = vecmax(sc); print1 (valuation(s,3)", "); sc = setunion(setminus(sc,[s]), Set([3*s/5, s/5]))) }

5^A356625(n) >= 3^a(n).

cp's OEIS Frontend

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

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A022337 Exponent of 5 (value of j) in n-th number of form 3^i*5^j.

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A112754 Total number of prime factors of n-th number of the form 3^i*5^j.

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A112761 Exponent of 3 (value of j) in n-th number of the form 2^i*3^j*5^k.

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A131579 Period 10: repeat 0, 3, 6, 9, 2, 5, 8, 1, 4, 7.

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A112753 Number of distinct prime factors of n-th number of the form 3^i*5^j.

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Author

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Crossrefs

Programs

Mathematica

Formula

A356624 After n iterations of the "Square Multiscale" substitution, the largest tiles have side length 3^t / 5^f; a(n) = t (A356625 gives corresponding f's).

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A112761 Exponent of 3 (value of j) in n-th number of the form 2^i3^j5^k.