A002473 -id:A002473 - cp's OEIS frontend

A068183 Largest number without decimal digits equal to 1 whose product of digits gives n!.

2, 32, 3222, 53222, 5332222, 75332222, 75332222222, 7533332222222, 755333322222222

Offset: 2

Labos Elemer, Feb 18 2002

Maximal solutions are obtained from the concatenation of distinct prime factors that have one decimal digit. The sequence is finite because, for n>10, n! has 2-digit prime factors.

Cf. A000142, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

a(n)=Max{x; f[x]=n!}, where x has no digit=1 and f[x_] := Apply[Times, IntegerDigits[x]].

A085124 Smallest multiple of n whose digital product is also a multiple of n, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 0, 168, 0, 378, 135, 48, 0, 36, 0, 0, 273, 0, 0, 168, 525, 0, 999, 476, 0, 0, 0, 288, 0, 0, 175, 1296, 0, 0, 0, 0, 0, 378, 0, 0, 495, 0, 0, 384, 3577, 0, 0, 0, 0, 1296, 0, 728, 0, 0, 0, 0, 0, 0, 1197, 448, 0, 0, 0, 0, 0, 0, 0, 1368, 0, 0, 3525, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy, Jul 06 2003

a(n) = 0 if n is not a term of A002473 (i.e., not a 7-smooth number).
a(n) = 0 if n is a multiple of 10.
Conjecture: otherwise, a(n) is positive.
Sequence may be redefined to avoid large number of zeros by restricting it to members of A002473 which are not divisible by 10.

Cf. A002473.

More terms from David Wasserman, Jan 27 2005

Comments edited by Jon E. Schoenfield, Mar 30 2014

A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245, 315, 343, 375, 405, 441, 525, 567, 625, 675, 729, 735, 875, 945, 1029, 1125, 1215, 1225, 1323, 1575, 1701, 1715, 1875, 2025, 2187, 2205, 2401, 2625, 2835, 3087

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005

nonn

The Heinz numbers of the partitions into parts 2,3, and 4 (including the number 1, the Heinz number of the empty partition). We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [2,3,3,4] the Heinz number is 3*5*5*7 = 525; it is in the sequence. - Emeric Deutsch , May 21 2015

Numbers m | 105^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A003586, A003591-A003595, A051037, A108319, A108513, A215366.

The odd terms of A002473.

[n: n in [1..4000] | PrimeDivisors(n) subset [3,5,7]]; // Bruno Berselli, Sep 24 2012

with(numtheory): S := {}: for j to 3100 do if `subset`(factorset(j), {3, 5, 7}) then S := `union`(S, {j}) else end if end do: S; # Emeric Deutsch, May 21 2015
# alternative
isA108347 := proc(n)
      if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {3, 5, 7} = {} );
    end if;
end proc:
A108347 := proc(n)
     option remember;
     if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA108347(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A108347(n),n=1..80); # R. J. Mathar, Jun 06 2024

With[{n = 3087}, Sort@ Flatten@ Table[3^i * 5^j * 7^k, {i, 0, Log[3, n]}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(3^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

from sympy import integer_log
def A108347(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):
                c -= integer_log(m//5**j,3)[0]+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - Amiram Eldar, Sep 22 2020

a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - Vaclav Kotesovec, Sep 23 2020

A234517 Sequence of numbers from A234515 such that A234515(n) > A234515(k) for all k < n.

Original entry on oeis.org

2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 480, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 12600, 15120, 20160, 27720, 30240, 32760, 55440, 65520, 83160, 110880, 131040, 166320, 196560, 221760, 277200, 332640, 360360, 393120, 415800, 443520

Offset: 1

Jaroslav Krizek, Jan 03 2014

nonn

A234515 = natural numbers n sorted by decreasing values of number k(n) = log_n(sigma(n)), where sigma(n) = A000203(n) = the sum of divisors of n.

Conjecture: sequence a(n) for n >= 4 is not the same as sequence of numbers from A234516 such that A234516(n) > A234516(k) for all k < n.

Cf. A002182 (numbers n such that tau(n) > tau(k) for all k < n), A002473 (numbers whose prime divisors are all <= 7).

Cf. A234515, A234516, A234518, A234519, A234520, A234521, A234522, A234523, A234524.

lista(nn) = {v = vector(nn, n, if (n==1, 0, log(sigma(n))/log(n))); v = vecsort(v,,5); m = 0; for (n=1, #v, if (v[n] > m, m = v[n]; print1(m, ", ")););} \\ Michel Marcus, Dec 11 2014

480 inserted and more terms from Michel Marcus, Dec 11 2014

A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 40, 42, 50, 60, 70, 80, 81, 90, 100, 105, 108, 112, 120, 140, 150, 160, 180, 200, 210, 240, 250, 270, 280, 300, 320, 350, 360, 400, 405, 420, 450, 480, 490, 500, 504, 540, 560, 600, 630, 640, 700, 720, 750, 800

Offset: 1

Daniel Mondot, Dec 18 2021

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 2.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for subsequent sequences A350181..., but not for this sequence (where mp(p(n)) = 1). That is because there are infinitely many numbers that include both an even digit (2, 4, 6 or 8), a 5 and no 0. For these numbers n, p(n) will include a zero and p(p(n)) will be 0.
Equivalently: This sequence contains all numbers A007954(k) such that A031346(k) = 2, and they are the numbers k in A002473 such that A031346(k) = 1.
Or, they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 1 step.

			10 is in this sequence because:
- 10 goes to a single digit in 1 step: p(10) = 0.
- 25, 52, 125, 152, 215, 512, 251, 521, 1125, 1152, 1215, 1512, 1251, 1521, 2115, 5112, 2511, 5211, etc. all lead to 10, i.e., p(25)=10, p(52)=10, etc.
Some of these (25, 125, 512, 1125, 1152, 1215, 1512) are in the next layer of classes, A350181, and the rest are not.
12 is in this sequence because:
- 12 goes to a single digit in 1 step: p(12) = 2.
- 12, 21, 112, 211, 121, 11112, 11211, etc. all lead to 12.
(12, 21 and 112 are in the next layer of classes, A350181, but the rest are not)
14 is in this sequence because:
- 14 goes to a single digit in 1 step: p(14) = 4.
- 27, 72, 127, 172, 217, 712, 271, 721, 12111711, etc. all lead to 14.
(27 and 72 are in the next layer of classes, A350181, the rest are not).

Daniel Mondot, Table of n, a(n) for n = 1..20000
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046510
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046510 (all numbers with mp of 1).
Cf. A350181, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 2 to 10 that are themselves 7-smooth numbers).

mp(n)={my(k=0); while(n>=10, k++; n=vecprod(digits(n))); k}
isparent(n)={my(m=0); while(m<>n, m=n; n/=gcd(n,2*3*5*7)); n==1}
isok(n)={mp(n)==1 && isparent(n)} \\ Andrew Howroyd, Dec 20 2021

A018336 Divisors of 210.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210

Offset: 1

N. J. A. Sloane

Also numbers that are squarefree and 7-smooth: intersection of A005117 and A002473. - Reinhard Zumkeller, Jul 15 2003

Index entries for sequences related to divisors of numbers

Divisors[210] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

divisors(210) \\ Charles R Greathouse IV, Jun 21 2017

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

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

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

A238985 Zeroless 7-smooth numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324, 336

Offset: 1

Charles R Greathouse IV and Reinhard Zumkeller, Mar 07 2014

A001221(a(n)) <= 3 since 10 cannot divide a(n).

It seems that this sequence is finite and contains 12615 terms. - Daniel Mondot, May 03 2022 and Jianing Song, Jan 28 2023

			a(12615) = 2^25 * 3^227 * 7^28.

Daniel Mondot, Table of n, a(n) for n = 1..12615 (terms 1..10000 from Charles R Greathouse IV)

Cf. A168046, intersection of A002473 and A052382.
A238938, A238939, A238940, A195948, A238936, A195908 are proper subsequences.
Cf. A059405 (subsequence), A350180 through A350187.

import Data.Set (singleton, deleteFindMin, fromList, union)
a238985 n = a238985_list !! (n-1)
a238985_list = filter ((== 1) . a168046) $ f $ singleton 1 where
   f s = x : f (s' `union` fromList
               (filter ((> 0) . (`mod` 10)) $ map (* x) [2,3,5,7]))
               where (x, s') = deleteFindMin s

zf(n)=vecmin(digits(n))
list(lim)=my(v=List(),t,t1); for(e=0,log(lim+1)\log(7), t1=7^e; for(f=0,log(lim\t1+1)\log(3), t=t1*3^f; while(t<=lim, if(zf(t), listput(v, t)); t<<=1)); for(f=0,log(lim\t1+1)\log(5), t=t1*5^f; while(t<=lim, if(zf(t), listput(v, t)); t*=3))); Set(v)

A086299(a(n)) * A168046(a(n)) = 1.

Keyword:fini and keyword:full removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

cp's OEIS Frontend

A068183 Largest number without decimal digits equal to 1 whose product of digits gives n!.

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A085124 Smallest multiple of n whose digital product is also a multiple of n, or 0 if no such number exists.

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A108347 Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.

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A234517 Sequence of numbers from A234515 such that A234515(n) > A234515(k) for all k < n.

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PARI

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A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

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PARI

A018336 Divisors of 210.

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Mathematica

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A085125 Even numbers which are 7-smooth.

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A085131 Multiples of 8 which are 7-smooth.

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Mathematica

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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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A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.