A071604 -id:A071604 - cp's OEIS frontend

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

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

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","); ) }

A085630 Number of n-digit 7-smooth numbers (A002473).

Original entry on oeis.org

0, 9, 36, 95, 197, 356, 579, 882, 1272, 1767, 2381, 3113, 3984, 5002, 6187, 7545, 9081, 10815, 12759, 14927, 17323, 19960, 22853, 26015, 29458, 33188, 37222, 41568, 46245, 51254, 56618, 62338, 68437, 74917, 81793, 89083, 96786, 104926, 113511

Offset: 0

Jason Earls and Amarnath Murthy, Jul 10 2003

Lars Blomberg, Table of n, a(n) for n = 0..100

Cf. A002473, A071604, A106600.

\\ Here b(n) is A071604.
b(m)={sum(i=0, logint(m,7), my(p=m\7^i); sum(j=0, logint(p,5), my(q=p\5^j); sum(k=0, logint(q,3), logint(q\3^k,2)+1 )))}
a(n)={if(n>0, b(10^n-1))-if(n>1, b(10^(n-1)-1))} \\ Andrew Howroyd, Sep 20 2024

From Andrew Howroyd, Sep 20 2024: (Start)
a(n) = A106600(n) - A106600(n-1) for n > 0.
a(n) = A071604(10^n-1) - A071604(10^(n-1)-1) for n > 1. (End)

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 18 2004

Name changed by Andrew Howroyd, Sep 20 2024

A096300 Number of positive integers <= n with no prime factor > log(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 1

Ralf Stephan, Aug 03 2004

A. Granville, On positive integers <= x with prime factors <= t log x, Number Theory and Applications (ed. R.A Mollin), (Kluwer, NATO ASI, 1989), 403-422.

a[n_] := Select[Range[n], FactorInteger[#][[-1, 1]] <= Log[n]&] // Length;
a[1] = a[2] = 1;
a /@ Range[75] (* Jean-François Alcover, May 17 2020 *)

a(n)=local(s, t); s=0; for(k=1, n, f=factor(k); t=0; for(l=1, matsize(f)[1], if(f[l, 1]>log(n), t=1; break)); s=s+!t); s

From Charlie Neder, Feb 08 2019: (Start)
a(n) = A000012(n) for 0 < n <= floor(e^2) = 7,
       A070939(n) for 7 < n <= floor(e^3) = 20,
       A071521(n) for 20 < n <= floor(e^5) = 148,
       A071520(n) for 148 < n <= floor(e^7) = 1096,
       A071604(n) for 1096 < n <= floor(e^11) = 59874,
and so on. (End)

cp's OEIS Frontend

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

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A080686 Number of 19-smooth numbers <= n.

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A071523 Number of 11-smooth numbers <= n.

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A080685 Number of 17-smooth numbers <= n.

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A085630 Number of n-digit 7-smooth numbers (A002473).

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A096300 Number of positive integers <= n with no prime factor > log(n).

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