A071520 -id:A071520 - cp's OEIS frontend

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

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

A080684 Number of 13-smooth numbers <= n.

Original entry on oeis.org

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

Offset: 1

Cino Hilliard, Mar 02 2003

Range = primes 2 to 13. Input pn=13 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.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A071520, A051038.

Accumulate[Table[If[Max[FactorInteger[n][[All,1]]]<14,1,0],{n,80}]] (* Harvey P. Dale, Jul 23 2018 *)

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 prevprime, integer_log
def A080684(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,13) # Chai Wah Wu, Oct 22 2024

A106598 Number of positive integers <= 10^n that are divisible by no prime exceeding 5.

Original entry on oeis.org

1, 9, 34, 86, 175, 313, 507, 768, 1105, 1530, 2053, 2683, 3429, 4301, 5310, 6466, 7780, 9259, 10917, 12761, 14801, 17048, 19511, 22201, 25127, 28300, 31730, 35425, 39397, 43654, 48207, 53066, 58243, 63746, 69584, 75769, 82310, 89216, 96499, 104168

Offset: 0

Robert G. Wilson v, May 27 2005

nonn

David A. Corneth, Table of n, a(n) for n = 0..1500

Row 3 of A253635.

Cf. A011557, A051037, A071520.

n = 40; t = Select[ Flatten[ Table[ 5^c*Select[ Flatten[ Table[2^a*3^b, {a, 0, n*Log[2, 10]}, {b, 0, n*Log[3, 10]}]], # <= 10^n &], {c, 0, n*Log[5, 10]}]], # <= 10^n &]; Table[ Length[ Select[t, # <= 10^n &]], {n, 0, 40}]

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","); ) }

A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 12 2003

T(n,n-1) = A014684(n) for n>1;
T(n,2) = A029837(n) for n>1; T(n,3) = A071521(n) for n>2; T(n,5) = A071520(n) for n>4.
A036234(n) = number of distinct terms in n-th row. - Reinhard Zumkeller, Sep 17 2013

			Triangle begins:
.................. 1
................ 1...2
.............. 1...2...3
............ 1...3...4...4
.......... 1...3...4...4...5
........ 1...3...5...5...6...6
...... 1...3...5...5...6...6...7
.... 1...4...6...6...7...7...8...8
.. 1...4...7...7...8...8...9...9...9.

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A002473, A003586, A006530, A014684, A029837, A036234, A051037, A051038, A071520, A071521, A080197, A080681, A080682, A080683.

a080786 n k = a080786_tabl !! (n-1) !! (k-1)
a080786_row n = a080786_tabl !! (n-1)
a080786_tabl = map reverse $ iterate f [1] where
   f xs@(x:_) = (x + 1) :
                (zipWith (+) xs (map (fromEnum . (lpf <=)) [x, x-1 ..]))
        where lpf = fromInteger $ a006530 $ fromIntegral (x + 1)
-- Reinhard Zumkeller, Sep 17 2013

A080786 := proc(x,y)
    local a,n ;
    a := 0 ;
    for n from 1 to x do
        if A006530(n) <= y then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 31 2013

P[n_] := FactorInteger[n][[-1, 1]]; P[1]=1; T[n_, k_] := (For[j=0; m=1, m <= n, m++, If[P[m] <= k, j++]]; j); Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2015 *)

from itertools import count, islice
from sympy import prevprime, integer_log
def A080786_T(n,k):
    if k==1: return 1
    def g(x,m): return x.bit_length() if m==2 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    return g(n,prevprime(k+1))
def A080786_gen(): # generator of terms
    return (A080786_T(n,k) for n in count(1) for k in range(1,n+1))
A080786_list = list(islice(A080786_gen(),100)) # Chai Wah Wu, Oct 22 2024

A112751 Number of numbers of the form 3^i*5^j that are less than or equal to n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A003593, A008683, A071520.

[&+[MoebiusMu(15*k)*Floor(n/k):k in [1..n]]: n in [1..97]]; // Marius A. Burtea, Jul 30 2019

with(numtheory): seq(add(mobius(15*k)*floor(n/k), k=1..n), n=1..90); # Ridouane Oudra, Jul 29 2019

Accumulate[Table[Boole[n == Times @@ ({3, 5}^IntegerExponent[n, {3, 5}])], {n, 1, 100}]] (* Amiram Eldar, May 04 2025 *)

From Ridouane Oudra, Jul 29 2019: (Start)
a(n) = Card_{ k | A003593(k) <= n }.
a(n) = Sum_{k=1..n} mu(15*k)*floor(n/k), where mu is the Möbius function (A008683).
a(n) = Sum_{k=1..n} (floor(15^k/k)-floor((15^k-1)/k)). (End)
From Ridouane Oudra, Jul 17 2020: (Start)
a(n) = Sum_{i=0..floor(log_5(n))} (floor(log_3(n/5^i)) + 1).
a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_5(n/3^i)) + 1). (End)

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)

A337881 30a(n) - 1 is the least prime of the form 2^r3^s*5^t - 1, r > 0, s > 0, t > 0, r + s + t = n.

Original entry on oeis.org

1, 2, 6, 8, 16, 48, 96, 432, 384, 512, 2304, 4608, 23040, 8192, 24576, 49152, 65536, 294912, 655360, 1310720, 2621440, 10616832, 6291456, 28311552, 62914560, 75497472, 251658240, 838860800, 402653184, 805306368, 1073741824, 12079595520, 14495514624, 65229815808

Offset: 3

Hugo Pfoertner and Rainer Rosenthal, Oct 12 2020

nonn

Hugo Pfoertner, Table of n, a(n) for n = 3..1000

Cf. A051037, A071520, A336773, A337882.

cp's OEIS Frontend

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

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A080686 Number of 19-smooth numbers <= n.

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A080684 Number of 13-smooth numbers <= n.

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A106598 Number of positive integers <= 10^n that are divisible by no prime exceeding 5.

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Mathematica

A071523 Number of 11-smooth numbers <= n.

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A080685 Number of 17-smooth numbers <= n.

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A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

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A112751 Number of numbers of the form 3^i*5^j that are less than or equal to n.

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A337881 30a(n) - 1 is the least prime of the form 2^r3^s*5^t - 1, r > 0, s > 0, t > 0, r + s + t = n.