A080686 -id:A080686 - 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

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

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","); ) }

A085269 Integer part of the conversion from centimeters to inches.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 30

Offset: 0

Cino Hilliard, Aug 12 2003

2.54 = 127/50. - Eric Desbiaux, Nov 16 2008

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, order 128.

Cf. A224233, A085270.

Floor[Range[0, 100]*100/254] (* Paolo Xausa, Jul 16 2025 *)

PARI
```
a(n) = floor(n*100/254);
```

a(n) = floor(n/2.54).

Eric Desbiaux suggested (Apr 19 2008) that A047332(n)-A080686(n) =? a(n-1), but R. J. Mathar points out that this is only true for the first 23 terms and so is nothing more than a coincidence. - N. J. A. Sloane, Apr 26 2008

Corrected by T. D. Noe, Nov 02 2006

A093700 Number of 9's immediately following the decimal point in the expansion of (3+sqrt(8))^n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 55, 56, 57

Offset: 1

Marvin Ray Burns, Apr 10 2004

Number of 0's immediately following the decimal point in the expansion of (3-sqrt(8))^n.

			Let n=10, (3+sqrt(8))^10= 45239073.9999999778... (the fractional part starts with seven 9's), so the 10th element in this sequence is 7.
The 132nd element is 100. The 1000th element is 765. The 1307th element is 1000.
The arrangement of repeating elements are like A074184 (Index of the smallest power of n >= n!) and A076539 (Numerators a(n) of fractions slowly converging to pi) and A080686 (Number of 19-smooth numbers <= n).

Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Experiment. Math. Volume 9, Issue 1 (2000), 3-12, Project Euclid - Cornell Univ (see Proposition 1).
Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Math Forum, Triangular Numbers That are Perfect Squares
Math Pages, On m = sqrt(sqrt(n) + sqrt(kn+1)) [Wrong link]
Robert Simms, Using counting numbers to generate Pythagorean triples

Cf. A003499, A050608, A080686, A076539, A074184.

For[n = 1, n < 999, n++, Block[{$MaxExtraPrecision = 50*n}, Print[ -Floor[Log[10, 1 - N[FractionalPart[(3 + 2Sqrt[2])^n], n]]] - 1]]]
f[n_] := Block[{}, -MantissaExponent[(3 - Sqrt[8])^n][[2]]]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Apr 10 2004 *)

Roughly, floor(3*n/4)

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

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A071604 a(n) is the number of 7-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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A085269 Integer part of the conversion from centimeters to inches.

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A093700 Number of 9's immediately following the decimal point in the expansion of (3+sqrt(8))^n.

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