A022337 -id:A022337 - cp's OEIS frontend

A112765 Exponent of highest power of 5 dividing n. Or, 5-adic valuation of n.

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

Offset: 1

Reinhard Zumkeller, Sep 18 2005

A027868 gives partial sums.

This is also the 5-adic valuation of Fibonacci(n). See Lengyel link. - Michel Marcus, May 06 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
T. Lengyel, The order of the Fibonacci and Lucas numbers, Fibonacci Quart. 33 (1995), no. 3, 234-239. See Lemma 1 p. 235.

Cf. A007814, A007949, A112762, A022337, A122840, A027868, A054899, A122841, A160093, A160094, A196563, A196564.

Cf. A343251, A000351 (positions of records, greedy inverse).

a112765 n = fives n 0 where
   fives n e | r > 0     = e
             | otherwise = fives n' (e + 1) where (n',r) = divMod n 5
-- Reinhard Zumkeller, Apr 08 2011

A112765 := proc(n)
    padic[ordp](n,5) ;
end proc: # R. J. Mathar, Jul 12 2016

a[n_] := IntegerExponent[n, 5]; Array[a, 105] (* Jean-François Alcover, Jan 25 2018 *)

A112765(n)=valuation(n,5); /* Joerg Arndt, Apr 08 2011 */

def a(n):
    k = 0
    while n > 0 and n%5 == 0: n //= 5; k += 1
    return k
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 06 2021

Totally additive with a(p) = 1 if p = 5, 0 otherwise.
From Hieronymus Fischer, Jun 08 2012: (Start)
With m = floor(log_5(n)), frac(x) = x-floor(x):
a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/5^j))).
a(n) = m + Sum_{j=1..m} (floor(-frac(n/5^j))).
a(n) = A027868(n) - A027868(n-1).
G.f.: Sum_{j>0} x^5^j/(1-x^5^j). (End)
a(5n) = A055457(n). - R. J. Mathar, Jul 17 2012
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/4. - Amiram Eldar, Feb 14 2021
a(n) = 5*Sum_{j=1..floor(log(n)/log(5))} frac(binomial(n, 5^j)*5^(j-1)/n). - Dario T. de Castro, Jul 10 2022

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

Original entry on oeis.org

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

A022336 Exponent of 3 (value of i) in n-th number of form 3^i*5^j.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 4, 0, 3, 2, 5, 1, 4, 0, 3, 6, 2, 5, 1, 4, 7, 0, 3, 6, 2, 5, 8, 1, 4, 7, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 10, 0, 3, 6, 9, 2, 5, 8, 11, 1, 4, 7, 10, 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 13, 0, 3, 6, 9, 12, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13, 0, 3, 6, 9, 12, 15, 2, 5, 8, 11, 14, 1, 4, 7

Offset: 1

Clark Kimberling

nonn

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

Cf. A003593, A007949, A022329, A022337, A112754, A112761.

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

a(n) = A007949(A003593(n)) = A112754(n) - A022337(n). - Reinhard Zumkeller, Sep 18 2005

A112762 Exponent of 5 (value of k) in n-th number of the form 2^i3^j5^k.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 2, 1, 0, 1, 0, 2, 0, 1, 3, 0, 1, 0, 2, 1, 0, 1, 0, 2, 0, 2, 1, 0, 3, 0, 1, 0, 2, 1, 0, 1, 3, 0, 2, 1, 0, 2, 1, 0, 3, 0, 1, 0, 2, 4, 1, 0, 2, 1, 0, 3, 0, 2, 1, 0, 2, 1, 0, 3, 0, 1, 3, 0, 2, 1, 4, 1, 0, 2, 1, 0, 3, 0, 2, 1, 0, 2, 4

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A022337, A051037, A112765, A112760, A112761.

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

a(n) = A112765(A051037(n));

a(n) = A112759(n) - A112760(n) - A112761(n).

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).

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.

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

Original entry on oeis.org

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

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     1          1/5
   5     4       81/625
   6     2         3/25
   7     5     243/3125
   8     3        9/125
   9     6    729/15625
  10     4       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. A022337, A329919, A354535, A356624.

{ sc = [1]; for (n=0, 76, s = vecmax(sc); print1 (-valuation(s,5)", "); sc = setunion(setminus(sc,[s]), Set([3*s/5, s/5]))) }

5^a(n) >= 3^A356624(n).

A173863 Square root of A172545(n).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Nov 26 2010

nonn

A054073(n+2). Essentially A022337.

cp's OEIS Frontend

A112765 Exponent of highest power of 5 dividing n. Or, 5-adic valuation of n.

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

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A022336 Exponent of 3 (value of i) in n-th number of form 3^i*5^j.

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A112762 Exponent of 5 (value of k) in n-th number of the form 2^i*3^j*5^k.

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

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Mathematica

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

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A173863 Square root of A172545(n).

A112762 Exponent of 5 (value of k) in n-th number of the form 2^i3^j5^k.