cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A080194 7-smooth numbers which are not 5-smooth.

Original entry on oeis.org

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 84, 98, 105, 112, 126, 140, 147, 168, 175, 189, 196, 210, 224, 245, 252, 280, 294, 315, 336, 343, 350, 378, 392, 420, 441, 448, 490, 504, 525, 560, 567, 588, 630, 672, 686, 700, 735, 756, 784, 840, 875, 882, 896, 945, 980
Offset: 1

Views

Author

Klaus Brockhaus, Feb 10 2003

Keywords

Comments

Numbers of the form 7*2^r*3^s*5^t*7^u with r, s, t, u >= 0.
Multiples of 7 which are members of A002473. Or multiples of 7 with the largest prime divisor < 10.
Numbers whose greatest prime factor (A006530) is 7. - M. F. Hasler, Nov 21 2018

Examples

			28 = 2^2*7 is a term but 30 = 2*3*5 is not.

Links

Crossrefs

Cf. A002473, A051037.
Cf. A085125, A085126, A085127, A085128, A085129, A085131, A085132.

Programs

  • Mathematica
    Select[Range[999], FactorInteger[#][[-1, 1]] == 7 &] (* Giovanni Resta, Nov 22 2018 *)
  • PARI
    A080194_list(M)={my(L=List(),a,b,c); for(r=1,logint(M\1,7), a=7^r; for(s=0, logint(M\a,3), b=a*3^s; for(t=0,logint(M\b,5), c=b*5^t; for(u=0,logint(M\c,2), listput(L,c<A051037. - Edited by M. F. Hasler, Nov 22 2018
  • PARI
    select( is_A080194(n)={n>1 && vecmax(factor(n,7)[,1])==7}, [0..10^3]) \\ Defines is_A080194(), used elsewhere. The select() command is a check and illustration. For longer lists, use list() above. - M. F. Hasler, Nov 21 2018
  • Python
    from sympy import integer_log
def A080194(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):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n)*7 # Chai Wah Wu, Sep 17 2024
  • Python
    # faster for initial segment of sequence
import heapq
from itertools import islice
from sympy import primerange
def A080194gen(p=7): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 7*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A080194gen(), 65))) # Michael S. Branicky, Sep 17 2024

Formula

a(n) = 7 * A002473(n). - David A. Corneth, Nov 22 2018
Sum_{n>=1} 1/a(n) = 5/8. - Amiram Eldar, Nov 10 2020

A085125 Even numbers which are 7-smooth.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 50, 54, 56, 60, 64, 70, 72, 80, 84, 90, 96, 98, 100, 108, 112, 120, 126, 128, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 210, 216, 224, 240, 250, 252, 256, 270, 280, 288, 294, 300, 320, 324, 336
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Comments

Equivalently, multiples of 2 with the largest prime divisor < 10.

Crossrefs

Intersection of A005843 and A002473.
Cf. A085126, A085127, A085128, A085129, A080194, A085131, A085132.

Programs

  • Mathematica
    Select[2*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Jul 06 2018 *)
  • Python
    from sympy import integer_log
def A085125(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()-1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 31 2025

Formula

From Amiram Eldar, Sep 23 2024: (Start)
a(n) = 2*A002473(n).
Sum_{n>=1} 1/a(n) = 35/16. (End)

Extensions

More terms from David Wasserman, Jan 28 2005
Offset changed by Andrew Howroyd, Sep 19 2024

A085126 Multiples of 3 which are members of A002473. Or multiples of 3 with the largest prime divisor < 10.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 36, 42, 45, 48, 54, 60, 63, 72, 75, 81, 84, 90, 96, 105, 108, 120, 126, 135, 144, 147, 150, 162, 168, 180, 189, 192, 210, 216, 225, 240, 243, 252, 270, 288, 294, 300, 315, 324, 336, 360, 375, 378, 384, 405, 420, 432, 441, 450
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Links

Crossrefs

Intersection of A008585 and A002473.
Cf. A085125, A085127, A085128, A085129, A080194, A085131, A085132.

Programs

  • Mathematica
    Select[3*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Apr 10 2019 *)
  • Python
    from sympy import integer_log
def A085126(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):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n)*3 # Chai Wah Wu, Sep 17 2024
  • Python
    # faster for initial segment of sequence
import heapq
from itertools import islice
def A085126gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 3*v
            oldv = v
            for p in psmooth_primes:
                    heapq.heappush(h, v*p)
print(list(islice(A085126gen(), 65))) # Michael S. Branicky, Sep 17 2024

Formula

a(n) = 3*A002473(n). - Chai Wah Wu, Sep 18 2024
Sum_{n>=1} 1/a(n) = 35/24. - Amiram Eldar, Sep 23 2024

Extensions

More terms from David Wasserman, Jan 28 2005
Offset changed to 1 by Michael S. Branicky, Sep 17 2024

A085127 Multiples of 4 which are members of A002473. Or multiples of 4 with the largest prime divisor < 10.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 48, 56, 60, 64, 72, 80, 84, 96, 100, 108, 112, 120, 128, 140, 144, 160, 168, 180, 192, 196, 200, 216, 224, 240, 252, 256, 280, 288, 300, 320, 324, 336, 360, 384, 392, 400, 420, 432, 448, 480, 500, 504, 512, 540, 560, 576
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Links

Crossrefs

Intersection of A008586 and A002473.
Cf. A085125, A085126, A085128, A085129, A080194, A085131, A085132.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
sort([seq(seq(seq(seq(2^a*3^b*5^c*7^d, d=0..floor(log[7](N/(2^a*3^b*5^c)))),c=0..floor(log[5](N/(2^a*3^b)))), b=0..floor(log[3](N/2^a))), a=2..floor(log[2](N)))]); # Robert Israel, Mar 18 2018
  • Mathematica
    Select[4Range[150],Last[FactorInteger[#]][[1]]<10&] (* Harvey P. Dale, Aug 24 2011 *)

Formula

a(n) = 4*A002473(n). - Robert Israel, Mar 18 2018
Sum_{n>=1} 1/a(n) = 35/32. - Amiram Eldar, Sep 23 2024

Extensions

More terms from David Wasserman, Jan 28 2005
Offset changed by Robert Israel, Mar 18 2018

A085131 Multiples of 8 which are 7-smooth.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 216, 224, 240, 256, 280, 288, 320, 336, 360, 384, 392, 400, 432, 448, 480, 504, 512, 560, 576, 600, 640, 648, 672, 720, 768, 784, 800, 840, 864, 896, 960, 1000, 1008, 1024
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Comments

Equivalently, multiples of 8 with the largest prime divisor < 10.

Links

Crossrefs

Intersection of A008590 and A002473.
Cf. A085125, A085126, A085127, A085128, A085129, A080194, A085132.

Programs

  • Mathematica
    Select[8*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Oct 22 2017 *)

Formula

From Amiram Eldar, Sep 22 2024: (Start)
a(n) = 8*A002473(n).
Sum_{n>=1} 1/a(n) = 35/64. (End)

Extensions

More terms from David Wasserman, Jan 28 2005
Offset changed by Andrew Howroyd, Sep 22 2024

A085128 Multiples of 5 which are members of A002473. Or multiples of 5 with the largest prime divisor <= 7.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 75, 80, 90, 100, 105, 120, 125, 135, 140, 150, 160, 175, 180, 200, 210, 225, 240, 245, 250, 270, 280, 300, 315, 320, 350, 360, 375, 400, 405, 420, 450, 480, 490, 500, 525, 540, 560, 600, 625, 630, 640, 675, 700
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Crossrefs

Intersection of A008587 (multiples of 5) and A002473 (7-smooth numbers).
Cf. A080194, A085125, A085126, A085127, A085129, A085131, A085132.

Programs

  • Mathematica
    With[{p = Prime[Range[4]]}, 5 * Select[Range[140], Times @@ (p^IntegerExponent[#, p]) == # &]] (* Amiram Eldar, Sep 22 2024 *)
  • PARI
    lista(nn) = {for (n=1, nn, if (vecmax(factor(5*n)[,1]) <= 7, print1(5*n, ", ")););} \\ Michel Marcus, Aug 15 2017

Formula

a(n) = 5*A002473(n). - Michel Marcus, Aug 15 2017
Sum_{n>=1} 1/a(n) = 7/8. - Amiram Eldar, Sep 22 2024

Extensions

More terms from David Wasserman, Jan 28 2005
Offset corrected by Michel Marcus, Aug 15 2017

A085132 Multiples of 9 which are 7-smooth.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 108, 126, 135, 144, 162, 180, 189, 216, 225, 243, 252, 270, 288, 315, 324, 360, 378, 405, 432, 441, 450, 486, 504, 540, 567, 576, 630, 648, 675, 720, 729, 756, 810, 864, 882, 900, 945, 972, 1008, 1080, 1125, 1134, 1152
Offset: 1

Views

Author

Amarnath Murthy, Jul 06 2003

Keywords

Comments

Equivalently, multiples of 9 with the largest prime divisor < 10.

Crossrefs

Intersection of A008591 and A002473.
Cf. A085125, A085126, A085127, A085128, A085129, A080194, A085131.

Programs

  • Mathematica
    With[{p = Prime[Range[4]]}, 9 * Select[Range[140], Times @@ (p^IntegerExponent[#, p]) == # &]] (* Amiram Eldar, Sep 23 2024 *)

Formula

From Amiram Eldar, Sep 22 2024: (Start)
a(n) = 9*A002473(n).
Sum_{n>=1} 1/a(n) = 35/72. (End)

Extensions

More terms from David Wasserman, Jan 28 2005
Offset changed by Andrew Howroyd, Sep 19 2024
Showing 1-7 of 7 results.

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