A238985 -id:A238985 - cp's OEIS frontend

A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192

Offset: 1

N. J. A. Sloane

Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182.
Successive numbers k such that phi(210k) = 48k. - Artur Jasinski, Nov 05 2008
The divisors of 10! (A161466) are a finite subsequence. - Reinhard Zumkeller, Jun 10 2009
Numbers n such that A198487(n) > 0 and A107698(n) > 0. - Jaroslav Krizek, Nov 04 2011
A262401(a(n)) = a(n). - Reinhard Zumkeller, Sep 25 2015
Numbers which are products of single-digit numbers. - N. J. A. Sloane, Jul 02 2017
Phi(a(n)) is 7-smooth. In fact, the Euler Phi function applied to p-smooth numbers, for any prime p, is p-smooth. - Richard Locke Peterson, May 09 2020
Also those integers k, such that, for every prime p > 5, p^(12k) - 1 == 0 (mod 5040k). - Federico Provvedi, Jun 06 2022
The nonprimes with this property are all terms except for 2, 3, 5 and 7, i.e.: (1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, ...); the composite terms are all but the first one of this subsequence. ["Trivial" data provided mainly for search purpose.] - M. F. Hasler, Jun 06 2023

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5841 terms from N. J. A. Sloane)
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.
University of Ulm, The first 5842 terms.
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Smooth number

Subsequence of A080672, complement of A068191. Subsequences: A003591, A003594, A003595, A195238, A059405.
Not the same as A063938. For p-smooth numbers with other values of p, see A003586, A051037, A051038, A080197, A080681, A080682, A080683.
Cf. A002182, A067374, A210679, A238985 (zeroless terms), A006530.
Cf. A262401.

import Data.Set (singleton, deleteFindMin, fromList, union)
a002473 n = a002473_list !! (n-1)
a002473_list = f $ singleton 1 where
   f s = x : f (s' `union` fromList (map (* x) [2,3,5,7]))
         where (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Mar 08 2014, Apr 02 2012, Apr 01 2012

[n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // Bruno Berselli, Sep 24 2012

Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]<=7&]
aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* Artur Jasinski, Nov 05 2008 *)
mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # <= 2^mxExp &] (* Harvey P. Dale, Aug 13 2012 *)
mx = 200; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,7, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A002473(n)=n<11||vecmax(factor(n,8)[,1])<8 \\ M. F. Hasler, Jan 16 2015

list(lim)=my(v=List(),t); for(a=0,logint(lim\1,7), for(b=0,logint(lim\7^a,5), for(c=0,logint(lim\7^a\5^b,3), t=3^c*5^b*7^a; while(t<=lim, listput(v,t); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017

import heapq
from itertools import islice
from sympy import primerange
def A002473gen(p=7): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A002473gen(), 65))) # Michael S. Branicky, Nov 19 2022

from sympy import integer_log
def A002473(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):
            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
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

A006530(a(n)) <= 7. - Reinhard Zumkeller, Apr 01 2012

Sum_{n>=1} 1/a(n) = Product_{primes p <= 7} p/(p-1) = (2*3*5*7)/(1*2*4*6) = 35/8. - Amiram Eldar, Sep 22 2020

More terms from James Sellers, Dec 23 1999
Additional comments from Michel Lecomte, Jun 09 2007
Edited by M. F. Hasler, Jan 16 2015

A059405 Numbers that are the product of their digits raised to positive integer powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 128, 135, 175, 384, 432, 672, 735, 1296, 1715, 6144, 6912, 13824, 18432, 23328, 34992, 82944, 93312, 131712, 248832, 442368, 1492992, 2239488, 2333772, 2612736, 3981312, 4128768, 4741632, 9289728, 12192768

Offset: 1

Erich Friedman, Jan 29 2001

The second example suggests that a repeated digit must divide the number at least as many times as it occurs, i.e., "distinct [digits]" in the definition would give a different (super)set. What would be the additional terms? - M. F. Hasler, Jan 05 2020

			a(17) = 1296 = (1)(2^2)(9)(6^2);
a(32) = 2333772 = (2)(3)(3)(3^3)(7)(7^3)(2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..120

Subsequence of A238985.

a059405 n = a059405_list !! (n-1)
a059405_list = filter f a238985_list where
   f x = all (== 0) (map (mod x) digs) && g x digs where
         g z []         = z == 1
         g z ds'@(d:ds) = r == 0 && (h z' ds' || g z' ds)
                          where (z', r) = divMod z d
         h z []         = z == 1
         h z ds'@(d:ds) = r == 0 && h z' ds' || g z ds
                          where (z', r) = divMod z d
         digs = map (read . return) $ filter (/= '1') $ show x
-- Reinhard Zumkeller, Apr 29 2015

More terms from Erich Friedman, Apr 01 2003

Offset changed by Reinhard Zumkeller, Apr 29 2015

A067183 Product of the prime factors of n equals the product of the digits of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 135, 175, 735, 1176, 1715, 131712

Offset: 1

Joseph L. Pe, Feb 18 2002

Terms are zeroless 7-smooth numbers (cf. A238985). - David A. Corneth, Sep 14 2022

			The prime factors of 1176 are 2,3,7 which have product = 42, the product of the digits of 1176, so 1176 is a term of the sequence.

Cf. A002473, A006753, A052382, A075048, A238985, A357132.

Do[ If[ Apply[ Times, Transpose[ FactorInteger[n]] [[1]] ] == Apply[ Times, IntegerDigits[n]], Print[n]], {n, 2, 2*10^7} ]
Select[Range[2,1000000],Times@@Transpose[FactorInteger[#]][[1]] == Times@@ IntegerDigits[#]&] (* Harvey P. Dale, Mar 19 2012 *)

is(n) = {if(n == 1, return(1)); my(f = factor(n, 7), d = digits(n)); if(f[#f~, 1] > 7, return(0)); vecprod(f[,1]) == vecprod(d)} \\ David A. Corneth, Sep 14 2022

Edited and extended by Robert G. Wilson v, Feb 19 2002

a(1)=1 inserted by Alois P. Heinz, Sep 14 2022

A357132 Numbers k such that the product of distinct digits of k equals the product of the prime divisors of k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 135, 175, 735, 1176, 1715, 13122, 131712, 2333772

Offset: 1

Alexandru Petrescu, Sep 14 2022

			175 = 5^2*7, 1*7*5 = 5*7. Thus 175 is a term.

Cf. A002473, A052382, A067183, A075048, A238985, A356981.

Select[Range[2400000], Times@@DeleteDuplicates[IntegerDigits[#]] == Times@@First/@FactorInteger[#] &] (* Stefano Spezia, Apr 25 2024 *)

isok(k) = vecprod(Set(digits(k))) == vecprod(factor(k)[, 1]);

A167620 Numbers that are multiples of their digital product, where this digital product also appears as their least significant digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 111, 112, 115, 315, 612, 1111, 1112, 1113, 1115, 1116, 11111, 11112, 11115, 12312, 13212, 21312, 23112, 31212, 32112, 111111, 111112, 111115, 111315, 111612, 113115, 116112, 131115, 161112, 311115, 511175

Offset: 1

Claudio Meller, Nov 07 2009

Subsequence of A007602. - R. J. Mathar, Nov 12 2009

The digital products of the terms are a subsequence of A238985. - Karl-Heinz Hofmann, Feb 16 2024

			612 is in the list because 6*1*2=12, 612 is a multiple of 12, and 12 is the final two digits of 612.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..8042 (Terms < 10^18, first 690 terms from David A. Corneth)

Cf. A007602, A238985, A194356.

is(n) = { my(vp = vecprod(digits(n))); vp != 0 && n %vp == 0 && n % 10^(#digits(vp)) == vp } \\ David A. Corneth, Mar 30 2021

A167620 = []
for k in range(1,511176):
    dprod, k_str = 1, str(k)
    for d in range(0,len(k_str)): dprod *= int(k_str[d])
    if dprod != 0 and k % dprod == 0 and str(dprod) == k_str[-(len(str(dprod))):]:
        A167620.append(k)
print(A167620) # Karl-Heinz Hofmann, Jan 26 2024

A359494 Zeroless numbers k which can be written as a product of the powers whose base is a digit of k and whose exponent is a nonnegative integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 32, 36, 64, 125, 128, 135, 175, 216, 243, 256, 324, 375, 384, 432, 512, 625, 672, 729, 735, 784, 864, 875, 1296, 1372, 1715, 1764, 1792, 2592, 2744, 2916, 3125, 3375, 3456, 3645, 3675, 4375, 5832, 6144, 6272, 6912, 7776, 8192, 8575, 9216

Offset: 1

Felix Huber, Jan 03 2023

As this sequence is a subsequence of A238985 which is conjectured to be finite, this sequence is conjectured to be finite. - David A. Corneth, Jan 28 2023

This sequence seems to contain 11055 terms. - Felix Huber, Apr 11 2024

			672 is in the sequence via 672 = 6^2 * 7^1 * 2^4.
2592 is in the sequence via 2592 = 2^5 * 5^0 * 9^2 * 2^0.

Felix Huber, Table of n, a(n) for n = 1..11055

Cf. A002473. Subsequence of A238985.

is(n) = {if(n <= 1, return(n == 1)); my(d = Set(digits(n))); if(d[1] == 0, return(0)); d = setminus(d, Set(1)); forvec(x = vector(#d, i, [0, valuation(n, d[i])]), c = prod(i = 1, #d, d[i]^x[i]); if(c == n, return(1) ) ); 0 } \\ David A. Corneth, Jan 05 2023

Integers k = (x_1)^(m_1) * ... * (x_i)^(m_i) where x_j is the j-th digit of zeroless k and m_j is a nonnegative integer and 1 <= j <= i.

A282782 Numbers that are equal to a product of powers of digits where the exponents from left to right decrease with 1 and the exponent for the units digit is 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1715

Offset: 1

Shmelev Aleksei, Feb 21 2017

Up to 10^9 no other number matches the rule.

There are no other terms up to 10^200; this list is probably complete. - Charles R Greathouse IV, Feb 22 2017

			1 = 1^1, 2 = 2^1, ..., 1715 = (1^4)*(7^3)*(1^2)*(5^1).
These numbers do not match the rule:
46: (4^2)*(6^1) = 96 <> 46.
234: (2^3)*(3^2)*(4^1) = 288 <> 234.
4342: (4^4)*(3^3)*(4^2)*(2^1) = 221184 <> 4342.
46914: (4^5)*(6^4)*(9^3)*(1^2)*(4^1) = 3869835264 <> 46914.

Subsequence of A002473; apart from the first term, a subsequence of A238985.

mx = 10^50; test[n_] := n == Times @@ (IntegerDigits[n] ^Reverse[Range@ IntegerLength@ n]); Union@Reap[Do[n = 2^i 3^j 7^k; If[test@n, Sow@n], {i, 0, Log2[mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[7, mx/2^i/3^j]}]; Do[n = 5  3^j 7^k; If[test@n, Sow@n], {j, 0, Log[3, mx/5]}, {k, 0, Log[7, mx/ 5/ 3^j]}]][[2, 1]] (* Search up to 10^50, Giovanni Resta, Feb 22 2017 *)
Select[Range[0, 2000], Times @@ MapIndexed[#1^First[#2] &, Reverse@ IntegerDigits@ #] == # &] (* Michael De Vlieger, Feb 22 2017 *)

is(n)=my(d=digits(n)); prod(i=1,#d,d[#d+1-i]^i)==n || !n \\ Charles R Greathouse IV, Feb 22 2017

list(lim)=my(v=List([0]),t7,t37,t); for(a=0,logint(lim\1,7), t7=7^a; for(b=0,logint(lim\t7,3), t=t37=t7*3^b; while(t<=lim, if(is(t), listput(v,t)); t<<=1); t=t37; while(t<=lim, if(is(t), listput(v,t)); t*=5))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017

' For example for 5-figure numbers:
Dim zahl As String
For i = 10000 To 99999
zahl = i
If i = CInt(Left(zahl, 1)) ^ 5 * CInt(Right(Left(zahl, 2), 1)) ^ 4 * CInt(Right(Left(zahl, 3), 1)) ^ 3 * CInt(Right(Left(zahl, 4), 1)) ^ 2 * CInt(Right(zahl, 1)) ^ 1 Then
MsgBox (i)
End If
Next i

Leading 0 prepended by David A. Corneth, Feb 22 2017

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A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.

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A059405 Numbers that are the product of their digits raised to positive integer powers.

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A067183 Product of the prime factors of n equals the product of the digits of n.

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A357132 Numbers k such that the product of distinct digits of k equals the product of the prime divisors of k.

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A167620 Numbers that are multiples of their digital product, where this digital product also appears as their least significant digits.

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A359494 Zeroless numbers k which can be written as a product of the powers whose base is a digit of k and whose exponent is a nonnegative integer.

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A282782 Numbers that are equal to a product of powers of digits where the exponents from left to right decrease with 1 and the exponent for the units digit is 1.

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