A080194 -id:A080194 - cp's OEIS frontend

A085130 Duplicate of A080194.

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 84, 98, 105, 112, 126, 140, 147, 168, 175, 189, 196

Offset: 1

dead

A085125 Even numbers which are 7-smooth.

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

Amarnath Murthy, Jul 06 2003

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

Intersection of A005843 and A002473.

Cf. A085126, A085127, A085128, A085129, A080194, A085131, A085132.

Select[2*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Jul 06 2018 *)

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

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

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

Amarnath Murthy, Jul 06 2003

Michael S. Branicky, Table of n, a(n) for n = 1..10001 (first 1001 terms from Harvey P. Dale)

Intersection of A008585 and A002473.

Cf. A085125, A085127, A085128, A085129, A080194, A085131, A085132.

Select[3*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Apr 10 2019 *)

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

# 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

a(n) = 3*A002473(n). - Chai Wah Wu, Sep 18 2024

Sum_{n>=1} 1/a(n) = 35/24. - Amiram Eldar, Sep 23 2024

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

Amarnath Murthy, Jul 06 2003

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A008586 and A002473.

Cf. A085125, A085126, A085128, A085129, A080194, A085131, A085132.

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

Select[4Range[150],Last[FactorInteger[#]][[1]]<10&] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = 4*A002473(n). - Robert Israel, Mar 18 2018

Sum_{n>=1} 1/a(n) = 35/32. - Amiram Eldar, Sep 23 2024

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

Amarnath Murthy, Jul 06 2003

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

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

Intersection of A008590 and A002473.

Cf. A085125, A085126, A085127, A085128, A085129, A080194, A085132.

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

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

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 22 2024

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 10, 9, 16, 11, 14, 15, 12, 32, 13, 22, 21, 20, 18, 64, 17, 26, 33, 28, 25, 24, 128, 19, 34, 39, 44, 35, 30, 27, 256, 23, 38, 51, 52, 55, 42, 40, 36, 512, 29, 46, 57, 68, 65, 66, 49, 45, 48, 1024, 31, 58, 69, 76, 85, 78, 77, 56, 50, 54, 2048, 37, 62, 87, 92

Offset: 1

Leroy Quet, Jan 27 2007

This sequence is a permutation of the integers >= 2.
Since the table has been entered by rising instead of falling antidiagonals, the sequence represents the transpose, with columns instead of rows: cf. the "table" link, section "infinite square array". - M. F. Hasler, Oct 22 2019
Start with table headed by primes in the first row, then list beneath each prime(k) the ordered prime(k)-smooth numbers. Read the table by falling antidiagonals to get the terms of this sequence. - David James Sycamore, Jun 23 2024

			Array begins: (rows here appear as columns in the "table" display of the sequence)
   2,  4,  8, 16, 32, 64, 128, 256, 512, ... (A000079)
   3,  6,  9, 12, 18, 24,  27,  36,  48, ... (A065119)
   5, 10, 15, 20, 25, 30,  40,  45,  50, ... (A080193)
   7, 14, 21, 28, 35, 42,  49,  56,  63, ... (A080194)
  11, 22, 33, 44, 55, 66,  77,  88,  99, ... (A080195)
  13, 26, 39, 52, 65, 78,  91, 104, 117, ... (A080196)
The 3rd row, for example, contains the positive integers where the 3rd prime, 5, is the largest prime divisor. That is, each integer in this row is divisible by 5 and may be divisible by 2 or 3 as well, but none of the integers in this row are divisible by primes larger than 5. (So for example, 35 = 5*7 is excluded from the 3rd row.)

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A083140, A006530, A000040 (1st col), A033286 (main diag), A077320.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681.

lpf[n_] := FactorInteger[n][[ -1, 1]];f[n_, m_] := f[n, m] = Block[{k},k = If[m == 1, Prime[n], f[n, m - 1] + 1];While[lpf[k] != Prime[n], k++ ];k];Table[f[ d - m + 1, m], {d, 12}, {m, d}] // Flatten (* Ray Chandler, Feb 09 2007 *)

T=List(); r=c=1; for(n=1,99, #TT[r][1], ); print1(T[r][c]","); r-- && c++ || r=c+c=1) \\ M. F. Hasler, Oct 22 2019

Extended by Ray Chandler, Feb 09 2007

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

Amarnath Murthy, Jul 06 2003

Intersection of A008587 (multiples of 5) and A002473 (7-smooth numbers).

Cf. A080194, A085125, A085126, A085127, A085129, A085131, A085132.

With[{p = Prime[Range[4]]}, 5 * Select[Range[140], Times @@ (p^IntegerExponent[#, p]) == # &]] (* Amiram Eldar, Sep 22 2024 *)

lista(nn) = {for (n=1, nn, if (vecmax(factor(5*n)[,1]) <= 7, print1(5*n, ", ")););} \\ Michel Marcus, Aug 15 2017

a(n) = 5*A002473(n). - Michel Marcus, Aug 15 2017

Sum_{n>=1} 1/a(n) = 7/8. - Amiram Eldar, Sep 22 2024

More terms from David Wasserman, Jan 28 2005

Offset corrected by Michel Marcus, Aug 15 2017

A085129 Multiples of 6 which are 7-smooth.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 84, 90, 96, 108, 120, 126, 144, 150, 162, 168, 180, 192, 210, 216, 240, 252, 270, 288, 294, 300, 324, 336, 360, 378, 384, 420, 432, 450, 480, 486, 504, 540, 576, 588, 600, 630, 648, 672, 720, 750, 756, 768, 810, 840

Offset: 1

Amarnath Murthy, Jul 06 2003

Equivalently, multiples of 6 with the largest prime divisor < 10.

Intersection of A008588 and A002473.

Cf. A085125, A085126, A085127, A085128, A080194, A085131, A085132.

With[{p = Prime[Range[4]]}, 6 * Select[Range[140], Times @@ (p^IntegerExponent[#, p]) == # &]] (* Amiram Eldar, Sep 23 2024 *)

From Amiram Eldar, Sep 22 2024: (Start)
a(n) = 6*A002473(n).
Sum_{n>=1} 1/a(n) = 35/48. (End)

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 19 2024

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

Amarnath Murthy, Jul 06 2003

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

Intersection of A008591 and A002473.

Cf. A085125, A085126, A085127, A085128, A085129, A080194, A085131.

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

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

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 19 2024

A219697 Primes neighboring a 7-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 107, 109, 113, 127, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 239, 241, 251, 257, 269, 271, 281, 293, 337, 349, 359, 379, 383, 401, 419, 421, 431

Offset: 1

Jonathan Vos Post, Nov 25 2012

This is to the 7-smooth numbers A002473 as A219528 is to the 3-smooth numbers A003586 and as A219669 is to the 5-smooth numbers A051037. The first primes NOT within one of a 7-smooth number are 103, 131, 137, 157, 173, ...

			23 is in the sequence as one of 23-1 = 22 = 2 * 11 and 23+1 = 24 = 2^3 * 3 is 7-smooth and 23 is prime. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10765 (terms <= 10^16)

Cf. A000040, A002473, A051037, A080194, A219528.

mx = 2^10; t7 = Select[Sort[Flatten[Table[2^i * 3^j * 5^k * 7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}, {l, 0, Log[7, mx]}]]], # <= mx &]; Union[Select[t7 + 1, PrimeQ], Select[t7 - 1, PrimeQ]] (* T. D. Noe, Nov 26 2012 *)
Select[Prime[Range[90]],Max[FactorInteger[#-1][[;;,1]]]<11||Max[FactorInteger[#+1][[;;,1]]]<11&] (* Harvey P. Dale, Nov 03 2024 *)

is7smooth(n) = forprime(p = 2, 7, n /= p^valuation(n, p)); n==1
is(n) = isprime(n) && (is7smooth(n - 1) || is7smooth(n + 1)) \\ David A. Corneth, Apr 19 2021

Primes INTERSECTION {2^h 3^i 5^j 7^k +/-1 for h,i,j,k >= 0}.

cp's OEIS Frontend

A085130 Duplicate of A080194.

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A085125 Even numbers which are 7-smooth.

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

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

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A085131 Multiples of 8 which are 7-smooth.

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A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

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

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A085129 Multiples of 6 which are 7-smooth.

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