A002473 -id:A002473 - cp's OEIS frontend

A068191 Numbers n such that A067734(n)=0; complement of A002473; at least one prime-factor of n is larger than 7, it has 2 decimal digits.

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102, 103, 104, 106, 107, 109, 110, 111, 113, 114

Offset: 1

Labos Elemer, Feb 19 2002

Also numbers n such that A198487(n) = 0 and A107698(n) = 0. - Jaroslav Krizek, Nov 04 2011
A086299(a(n)) = 0. - Reinhard Zumkeller, Apr 01 2012
A262401(a(n)) < a(n). - Reinhard Zumkeller, Sep 25 2015
Numbers not in A007954. - Mohammed Yaseen, Sep 13 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A002473, A067734.
Cf. A068183, A068184, A068185, A068186, A068187, A068189, A068190.
Cf. A262401.

import Data.List (elemIndices)
a068191 n = a068191_list !! (n-1)
a068191_list = map (+ 1) $ elemIndices 0 a086299_list
-- Reinhard Zumkeller, Apr 01 2012

Select[Range@120, Last@Map[First, FactorInteger@#] > 7 &] (* Vincenzo Librandi, Sep 19 2016 *)

from sympy import integer_log
def A068191(n):
    def f(x):
        c = n
        for i in range(integer_log(x,7)[0]+1):
            i7 = 7**i
            m = x//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
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 16 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

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

A326318 Numbers that cannot be written as a difference of 7-smooth numbers (A002473).

Original entry on oeis.org

1849, 2309, 2411, 2483, 2507, 2531, 2629, 2711, 2753, 2843, 2851, 2921, 2941, 3139, 3161, 3167, 3181, 3217, 3229, 3251, 3287, 3289, 3293, 3323, 3379, 3481, 3487, 3541, 3601, 3623, 3653, 3697, 3698, 3709, 3737, 3739, 3803, 3827, 3859, 3877, 3901, 3923, 3947

Offset: 1

Esteban Crespi de Valldaura, Jun 26 2019

nonn

Terms were found by generating in sequential order the 7-smooth numbers up to some limit and collecting the differences. The first 100 candidates k were then proved to be correct by showing that each of the following congruences holds:
   <2> +- k !== <3, 5, 7> mod 31487336959,
   <3> +- k !== <2, 5, 7> mod 121328339431,
   <2, 3> +- k !== <5, 7> mod 5699207989579,
   <5> +- k !== <2, 3, 7> mod 1206047658673,
   <2, 5> +- k !== <3, 7> mod 11174958041,
   <3, 5> +- k !== <2, 7> mod 31487336959,
   <7> +- k !== <2, 3, 5> mod 1116870318707,
where  represents any element in the subgroup generated by a,b,... of the multiplicative subgroup modulo m. For a discussion iof this method of proof see A308247.

			1849 = A308247(4) cannot be written as the difference of 7-smooth numbers. All smaller numbers can; for example, 281 = 2^5*3^2 - 7, 289 = 2*3*7^2 - 5, ..., 1847 = 3*5^4 - 2^2*7.

Esteban Crespi de Valldaura, Table of n, a(n) for n = 1..101

Cf. A002473 (7-smooth numbers).
Cf. numbers not the difference of p-smooth numbers for other values of p: A101082 (p=2), A290365 (p=3), A308456 (p=5), A326319 (p=11), A326320 (p=13).
Cf. A308247.

A372401 Position of 210^n among 7-smooth numbers A002473.

Original entry on oeis.org

1, 68, 547, 2119, 5817, 13008, 25412, 45078, 74409, 116147, 173379, 249532, 348375, 474018, 630922, 823885, 1058051, 1338898, 1672260, 2064302, 2521535, 3050825, 3659361, 4354687, 5144682, 6037582, 7041946, 8166692, 9421074, 10814695, 12357491, 14059744, 15932086, 17985473

Offset: 0

Michael De Vlieger, Jun 03 2024

nonn

Also position of 210^(n+1) in A147571.

Cf. A002110, A002473, A022330, A147571, A202821, A372400, A372402.

Table[
  Sum[Floor@ Log[7, 210^n/(2^i*3^j*5^k)] + 1,
    {i, 0, Log[2, 210^n]},
    {j, 0, Log[3, 210^n/2^i]},
    {k, 0, Log[5, 210^n/(2^i*3^j)]}],
  {n, 0, 12}]

import heapq
from itertools import islice
from sympy import primerange
def A372401gen(p=7): # generator for p-smooth terms
    v, oldv, psmooth_primes, = 1, 0, list(primerange(1, p+1))
    h = [(1, [0]*len(psmooth_primes))]
    idx = {psmooth_primes[i]:i for i in range(len(psmooth_primes))}
    loc = 0
    while True:
        v, e = heapq.heappop(h)
        if v != oldv:
            loc += 1
            if len(set(e)) == 1:
                yield loc
            oldv = v
            for p in psmooth_primes:
                vp, ep = v*p, e[:]
                ep[idx[p]] += 1
                heapq.heappush(h, (v*p, ep))
print(list(islice(A372401gen(), 15))) # Michael S. Branicky, Jun 05 2024

from sympy import integer_log
def A372401(n):
    c, x = 0, 210**n
    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 # Chai Wah Wu, Sep 16 2024

a(n) ~ c * n^4, where c = log(210)^4/(24*log(2)*log(3)*log(5)*log(7)) = 14.282278766622... - Vaclav Kotesovec and Amiram Eldar, Sep 22 2024

A280249 a(n) = smallest k with digit product A002473(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 35, 28, 29, 45, 37, 38, 55, 39, 47, 56, 48, 57, 49, 58, 67, 59, 68, 77, 255, 69, 78, 256, 79, 88, 257, 89, 355, 258, 99, 267, 259, 268, 277, 455, 357, 269, 278, 358, 555, 279, 288, 359, 457, 289, 377, 556, 458, 299, 378, 557, 459, 379, 388, 477, 558, 567, 389, 478, 559, 568, 399, 577, 2555, 479

Offset: 1

David W. Wilson, Dec 29 2016

A007954(a(n)) = A002473(n).

David W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A002473, A007954.

A323050 Numbers that cannot be written as a sum of two or fewer 7-smooth numbers (A002473).

Original entry on oeis.org

311, 479, 551, 619, 622, 671, 719, 839, 851, 933, 937, 958, 1102, 1103, 1117, 1151, 1193, 1238, 1244, 1291, 1319, 1342, 1391, 1433, 1437, 1438, 1487, 1499, 1511, 1531, 1553, 1555, 1559, 1619, 1651, 1653, 1667, 1678, 1679, 1697, 1857, 1866, 1871, 1874, 1913, 1916, 1919, 1933, 1937, 1991, 2011, 2013, 2077, 2113, 2117, 2157

Offset: 1

Carlos Alves, Jan 03 2019

nonn

Numbers that are not of the form (2^i * 3^j * 5^k * 7^l)*a + (2^m * 3^n * 5^p * 7^q)*b, with i,j,k,m,n,p >= 0, and a,b = 0 or 1. The first number excluded is 311.

These numbers are also included in A323046 and A323049.

Similar to A323046 (for 3-smooth) and A323049 (for 5-smooth). Cf. A002473.

f[n_] := Union@Flatten@Table[2^a*3^b*5^c*7^d, {a, 0, Log2[n]}, {b, 0, Log[3, n/2^a]}, {c, 0, Log[5, n/(2^a*3^b)]}, {d, 0, Log[7, n/(2^a*3^b*5^c)]}];
b = Block[{nn = 3000, s}, s = f[nn]; {0, 1}~Join~Select[Union@Flatten@Outer[Plus, s, s], # <= nn &]];
Complement[Range[3000], b]

A085133 Numbers k such that k and its digit reversal are both 7-smooth (A002473).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 120, 144, 180, 200, 210, 240, 252, 270, 288, 300, 343, 360, 400, 405, 420, 441, 450, 480, 500, 504, 525, 540, 576, 600, 630, 675, 686, 700, 720

Offset: 1

Amarnath Murthy, Jul 06 2003

Though a large number of initial terms match, it is different from A005349.
From Robert Israel, Mar 18 2018: (Start)
If n is a term, then so are 10^k*n for all k.
Is a(147)=84672 the last term not divisible by 10?  If so, then a(n+43)=10*a(n) for n >= 105. (End)
All terms a(147..10000) are divisible by 10; a(10000) has 235 decimal digits. - Michael S. Branicky, Sep 21 2024

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1225 from Robert Israel)

Cf. A005349, A002473.

N:= 10^3: # to get all terms <= N (which should be a power of 10)
revdigs:= proc(n) local L;
  L:= convert(n,base,10);
  add(10^(i-1)*L[-i],i=1..nops(L))
end proc:
S:= {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=0..floor(log[2](N)))}:
S:= S intersect map(revdigs, S):
S:= map(t -> seq(t*10^i, i=0..ilog10(N/t)), S):
sort(convert(S,list)); # Robert Israel, Mar 18 2018

import heapq
from itertools import islice
from sympy import factorint
def is7smooth(n):
    for p in [2, 3, 5, 7]:
        while n%p == 0: n //= p
    return n == 1
def agen(): # generator of terms
    v, oldv, h = 1, 0, [1]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            if is7smooth(int(str(v)[::-1])):
                yield v
            oldv = v
            for p in [2, 3, 5, 7]:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 65))) # Michael S. Branicky, Sep 20 2024

More terms from David Wasserman, Jan 28 2005

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

cp's OEIS Frontend

A068191 Numbers n such that A067734(n)=0; complement of A002473; at least one prime-factor of n is larger than 7, it has 2 decimal digits.

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

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A326318 Numbers that cannot be written as a difference of 7-smooth numbers (A002473).

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A372401 Position of 210^n among 7-smooth numbers A002473.

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A280249 a(n) = smallest k with digit product A002473(n).

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A323050 Numbers that cannot be written as a sum of two or fewer 7-smooth numbers (A002473).

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A085133 Numbers k such that k and its digit reversal are both 7-smooth (A002473).