A071521 -id:A071521 - cp's OEIS frontend

A065331 Largest 3-smooth divisor of n.

1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 27, 4, 1, 6, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 12, 1, 2, 9, 64, 1, 6, 1, 4, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 81, 2, 1, 12, 1, 2, 3, 8, 1, 18, 1, 4, 3, 2, 1, 96

Offset: 1

Reinhard Zumkeller, Oct 29 2001

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) < n^0.715. - Charles R Greathouse IV, May 21 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. A. Bennett, M. Filaseta, and O. Trifonov, On the factorization of consecutive integers, J. Reine Angew. Math. 629 (2009), pp. 171-200.
Jon Maiga, Computer-generated formulas for A065331, Sequence Machine.

Cf. A003586, A065330, A006519, A038500, A007814, A007949, A007310, A047229, A071521, A322026.
Related to A053165 via A225546.
Cf. A126760 (ordinal transform of this sequence, from its term a(1) = 1 onward).

a065331 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 3 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

[Gcd(n,6^n): n in [1..100]]; // Vincenzo Librandi, Feb 09 2016

A065331 := proc(n) n/A065330(n) ; end: # R. J. Mathar, Jun 24 2009
seq(2^padic:-ordp(n,2)*3^padic:-ordp(n,3), n=1..100); # Robert Israel, Feb 08 2016

Table[GCD[n, 6^n], {n, 100}] (* Vincenzo Librandi, Feb 09 2016 *)
a[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

a(n)=gcd(n,6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016

a(n)=gcd(6^logint(n,2),n) \\ 'optimized' version of Sykora's script; Charles R Greathouse IV, Jul 23 2024

a(n) = n / A065330(n).
a(n) = A006519(n) * A038500(n).
a(n) = (2^A007814 (n)) * (3^A007949(n)).
Multiplicative with a(2^e)=2^e, a(3^e)=3^e, a(p^e)=1, p>3. - Vladeta Jovovic, Nov 05 2001
Dirichlet g.f.: zeta(s)*(1-2^(-s))*(1-3^(-s))/ ( (1-2^(1-s))*(1-3^(1-s)) ). - R. J. Mathar, Jul 04 2011
a(n) = gcd(n,6^n). - Stanislav Sykora, Feb 08 2016
a(A225546(n)) = A225546(A053165(n)). - Peter Munn, Jan 17 2020
Sum_{k=1..n} a(k) ~ n*(log(n)^2 + (2*gamma + 3*log(2) + 2*log(3) - 2)*log(n) + (2 + log(2)^2/6 + 3*log(2)*(log(3) - 1) - 2*log(3) + log(3)^2/6 + gamma*(3*log(2) + 2*log(3) - 2) - 2*sg1)) / (6*log(2)*log(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Sep 19 2020
a(n) = A003586(A322026(n)), A322026(n) = A071521(a(n)). - Antti Karttunen, Sep 08 2024

A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 29 2001

Dirichlet inverse of b(n) where b(n) = 0 except for: b(1) = b(6) = -b(2) = -b(3) = 1. - Alexander Adam, Dec 26 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 9.
Index entries for characteristic functions

Characteristic function of A003586.

Cf. A000265, A007814, A007949, A038502, A065330, A065332, A071521 (partial sums), A072078 (inverse Möbius transform).

a065333 = fromEnum . (== 1) . a038502 . a000265
-- Reinhard Zumkeller, Jan 08 2013, Apr 12 2012

a[n_] := Boole[ 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] == n]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 16 2013, after Charles R Greathouse IV *)

a(n)=sumdiv(n,d,moebius(6*d)) \\ Benoit Cloitre, Oct 18 2009

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

from sympy import multiplicity
def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m) # Chai Wah Wu, Dec 20 2024

a(n) = if n = A003586(k) for some k then 1 else 0.
a(n) = signum(A065332(n)), where signum = A057427.
a(n) = if A065330(n) = 1 then 1 else 0 = 1 - signum(A065330(n) - 1).
a(n) = Product_{p prime and p|n} 0^floor(p/4). - Reinhard Zumkeller, Nov 19 2004
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = 0 for prime p > 3. Dirichlet g.f. 1/(1-2^-s)/(1-3^-s). - Franklin T. Adams-Watters, Sep 01 2006
a(n) = 0^(A038502(A000265(n)) - 1). - Reinhard Zumkeller, Sep 28 2008
a(n) = Sum_{d|n} mu(6*d). - Benoit Cloitre, Oct 18 2009

A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A122841(i) = A122841(j),
  a(i) = a(j) => A244417(i) = A244417(j),
  a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).
If and only if a(k) > a(i) for all k > i then k is in A003586, - David A. Corneth, Dec 03 2018
That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.
Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.

Cf. also A247714 and A255975.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];

A065331(n) = (3^valuation(n, 3)<A065331
A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
A322026(n) = A071521(A065331(n)); \\ Antti Karttunen, Sep 08 2024

For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - David A. Corneth, Dec 03 2018
A065331(n) = A003586(a(n)). - David A. Corneth, Dec 04 2018
From Antti Karttunen, Sep 08 2024: (Start)
a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.
a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]
a(n) = A323884(25*n). [Conjectured by Sequence Machine]
(End)

A022330 Index of 3^n within sequence of numbers of form 2^i*3^j (A003586).

Original entry on oeis.org

1, 3, 7, 12, 19, 27, 37, 49, 62, 77, 93, 111, 131, 152, 175, 199, 225, 252, 281, 312, 344, 378, 413, 450, 489, 529, 571, 614, 659, 705, 753, 803, 854, 907, 961, 1017, 1075, 1134, 1195, 1257, 1321, 1386, 1453, 1522, 1592, 1664, 1737, 1812, 1889, 1967, 2047, 2128

Offset: 0

Clark Kimberling

nonn

a(1000)=793775, a(10000)=79261054, a(100000)=7924941755, a(1000000)=792482542841.

Zak Seidov, Table of n, a(n) for n = 0..10000 (terms for n = 0..1000 from Charles R Greathouse IV).
N. Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, arXiv preprint arXiv:1303.0888 [math.CO], 2013.
N. Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, J. Int. Seq. 16 (2013) #13.3.4.

Cf. A022331, A020914 (first differences).

Cf. A000244, A003586, A071521.

c[0] = 1; c[n_] := 1 + Sum[Ceiling[j*Log[2, 3]], {j, n}]; Table[c[i], {i, 0, 51}] (* Norman Carey, Jun 13 2012 *)

listsm(lim)=my(v=List(),N); for(n=0,log(lim)\log(3),N=3^n; while(N<=lim,listput(v,N);N<<=1)); v=Vec(v); vecsort(v)
list(lim)=my(v=listsm(3^floor(lim)));vector(floor(lim+1),i,setsearch(v,3^(i-1))) \\ Charles R Greathouse IV, Aug 19 2011

a(n)=sum(k=0,n, logint(3^k,2))+n+1 \\ Charles R Greathouse IV, Nov 22 2022

def A022330(n): return sum((3**i).bit_length() for i in range(n+1)) # Chai Wah Wu, Sep 16 2024

a(n) = A071521(A000244(n)); A003586(a(n)) = A000244(n). - Reinhard Zumkeller, May 09 2006
a(n) ~ kn^2 with k = log(3)/log(4) = 0.792.... More exact asymptotics? - Zak Seidov, Dec 22 2011
a(n+1) = a(n) + A020914(n+1). - Ruud H.G. van Tol, Nov 25 2022
kn^2 + kn + 1 <= a(n) <= kn^2 + (k+1)n + 1, so a(n) = kn^2 + O(n) with k = log(3)/log(4). The law of the iterated logarithm suggests that a better error term might be possible. - Charles R Greathouse IV, Nov 28 2022

A071520 Number of 5-smooth numbers (A051037) <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 28

Offset: 1

Benoit Cloitre, Jun 02 2002

A 5-smooth number is a number of the form 2^x*3^y*5^z (x,y,z) >= 0.

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

Cf. A008683, A051037, A106598, A112751.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), this sequence (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).

Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<6,1,0],{n,80}]] (* Harvey P. Dale, Aug 04 2024 *)

for(n=1,100,print1(sum(k=1,n,if(sum(i=4,n,if(k%prime(i),0,1)),0,1)),","))

a(n)=-sum(k=1,n,moebius(2*3*5*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

from sympy import integer_log
def A071520(n):
    c = 0
    for i in range(integer_log(n,5)[0]+1):
        for j in range(integer_log(m:=n//5**i,3)[0]+1):
            c += (m//3**j).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A051037(k) <= n }.
Asymptotically : let a = 1/(6*log(2)*log(3)*log(5)) and b = sqrt(30) then a(n) = a*log(b*n)^3 + O(log(n)).
a(n) = -Sum_{k=1,n} mu(30*k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{i=0..floor(log_5(n))} Sum_{j=0..floor(log_3(n/5^i))} floor(log_2(2*n/(5^i*3^j))). - Ridouane Oudra, Jul 17 2020

Title corrected by Rainer Rosenthal, Aug 30 2020

A022331 Index of 2^n within sequence of numbers of form 2^i*3^j (A003586).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 22, 28, 34, 41, 48, 56, 65, 74, 84, 95, 106, 118, 130, 143, 157, 171, 186, 202, 218, 235, 253, 271, 290, 309, 329, 350, 371, 393, 416, 439, 463, 487, 512, 538, 564, 591, 619, 647, 676, 706, 736, 767, 798, 830, 863, 896, 930, 965, 1000, 1036, 1072

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)
Norman Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, arXiv preprint arXiv:1303.0888 [math.CO], 2013; Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, J. Int. Seq. 16 (2013), Article 13.3.4.

Cf. A000079, A003586, A071521, A020915 (first differences), A152747.

Cf. A022330 (index of 3^n within A003586).

c[0] = 1; c[n_] := 1 + Sum[Ceiling[j*Log[3, 2]], {j, n}]; Table[c[i], {i, 0, 60}] (* Norman Carey, Jun 13 2012 *)

a(n)=my(t=1);1+n+sum(k=1,n,logint(t*=2,3)) \\ Ruud H.G. van Tol, Nov 25 2022

from sympy import integer_log
def A022331(n):
    m = 1<Chai Wah Wu, Sep 16 2024

a(n) = A071521(A000079(n)); A003586(a(n)) = A000079(n). - Reinhard Zumkeller, May 09 2006

a(n) ~ c * n^2, where c = log(2)/(2*log(3)) (A152747). - Amiram Eldar, Apr 07 2023

A071604 a(n) is the number of 7-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 36, 36, 37, 37, 38

Offset: 1

Benoit Cloitre, Jun 02 2002

A 7-smooth number is a number of the form 2^x*3^y*5^z*7^u, (x,y,z,u) >= 0.

In other words, a 7-smooth number is a number with no prime factor greater than 7. - Peter Munn, Nov 20 2021

			a(11) = 10 as there are 10 7-smooth numbers <= 11. Namely 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Partial sums of A086299.
Column 7 of A080786.
Cf. A002473, A106600, A343598.
Equivalent sequences with other limits on greatest prime factor: A070939 (2), A071521 (3), A071520 (5), A071523 (11), A080684 (13), A080685 (17), A080686 (19), A096300 (log n).

for(n=1,100,print1(sum(k=1,n,if(sum(i=5,n,if(k%prime(i),0,1)),0,1)),","))

from sympy import integer_log
def A071604(n):
    c = 0
    for i in range(integer_log(n,7)[0]+1):
        i7 = 7**i
        m = n//i7
        for j in range(integer_log(m,5)[0]+1):
            j5 = 5**j
            r = m//j5
            for k in range(integer_log(r,3)[0]+1):
                c += (r//3**k).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A002473 (k) <= n }.

Name corrected by David A. Corneth, Apr 19 2021

A080686 Number of 19-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37, 38, 39, 39, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 48, 48, 49, 49, 49, 50, 51, 52, 53, 53, 54, 54, 55, 55, 56

Offset: 1

Cino Hilliard, Mar 02 2003

Range = primes 2 to 19. Input pn=19 in script below. Code below is much faster than the code for cross-reference. For input of n=200 13 times as fast and many times faster for larger input of n.

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

Cf. A080682.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), A071520 (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), A080685 (p=17), this sequence (p=19).

Accumulate[Table[Boole[Max[FactorInteger[n][[;; , 1]]] <= 19], {n, 100}]] (* Amiram Eldar, Apr 29 2025 *)

smoothn(n,pn) = { for(m=1,n, pr=1; forprime(p=2,pn, pr*=p; ); ct=1; for(x=1,m, f=0; forprime(y=nextprime(pn+1),floor(x), if(x%y == 0,f=1; break) ); if(gcd(x,pr)<>1,if(f==0,ct+=1; )) ); print1(ct","); ) }

from sympy import integer_log, prevprime
def A080686(n):
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    return g(n,19) # Chai Wah Wu, Sep 17 2024

A071523 Number of 11-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 28, 28, 29, 29, 30, 30, 31, 32, 32, 32, 33, 34, 35, 35, 35, 35, 36, 37, 38, 38, 38, 38, 39, 39, 39, 40, 41, 41, 42, 42, 42, 42, 43, 43, 44

Offset: 1

Benoit Cloitre, Jun 02 2002

An 11-smooth number is a number of the form 2^x*3^y*5^z*7^u*11^v (x,y,z,u,v) >= 0.

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

Cf. A051038.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), A071520 (p=5), A071604 (p=7), this sequence (p=11), A080684 (p=13), A080685 (p=17), A080686 (p=19).

Accumulate[Table[If[Max[FactorInteger[n][[;;,1]]]<=11,1,0],{n,120}]] (* Harvey P. Dale, Sep 02 2024 *)

a(n)=sum(k=1,n,(k<4) || 13>vecmax(factor(k)~[1,]))

a(n) = Card{ k | A051038(k) <= n }.

A080685 Number of 17-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 31, 32, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 45, 45, 46, 46, 46, 47, 48, 49, 50, 50, 51, 51, 52, 52, 53

Offset: 1

Cino Hilliard, Mar 02 2003

Range = primes 2 to 17. Input pn=17 in script below. Code below is much faster than the code for cross-reference. For input of n=200 13 times as fast and many times faster for larger input of n.

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

Cf. A080681.

Number of p-smooth numbers <= n: A070939 (p=2), A071521 (p=3), A071520 (p=5), A071604 (p=7), A071523 (p=11), A080684 (p=13), this sequence (p=17), A080686 (p=19).

Accumulate[Table[Boole[Max[FactorInteger[n][[;; , 1]]] <= 17], {n, 100}]] (* Amiram Eldar, Apr 29 2025 *)

smoothn(n,pn) = { for(m=1,n, pr=1; forprime(p=2,pn, pr*=p; ); ct=1; for(x=1,m, f=0; forprime(y=nextprime(pn+1),floor(x), if(x%y == 0,f=1; break) ); if(gcd(x,pr)<>1,if(f==0,ct+=1; )) ); print1(ct","); ) }

cp's OEIS Frontend

A065331 Largest 3-smooth divisor of n.

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A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

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A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

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A022330 Index of 3^n within sequence of numbers of form 2^i*3^j (A003586).

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A071520 Number of 5-smooth numbers (A051037) <= n.

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A022331 Index of 2^n within sequence of numbers of form 2^i*3^j (A003586).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula