A061427 -id:A061427 - cp's OEIS frontend

A174813 a(n) = number whose product of digits equals a power of 3.

1, 3, 9, 11, 13, 19, 31, 33, 39, 91, 93, 99, 111, 113, 119, 131, 133, 139, 191, 193, 199, 311, 313, 319, 331, 333, 339, 391, 393, 399, 911, 913, 919, 931, 933, 939, 991, 993, 999, 1111, 1113, 1119, 1131, 1133, 1139, 1191, 1193, 1199, 1311, 1313, 1319, 1331

Offset: 1

Michel Lagneau, Dec 01 2010

Equivalently, numbers whose decimal representation consists of digits from the set {1,3,9}.

			a(9)=39 is in the sequence because 3*9=3^3.

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

Cf. A028846, A061427.

a174813 n = a174813_list !! (n-1)
a174813_list = f [1] where
   f ds = foldr (\d v -> 10 * v + d) 0 ds : f (s ds)
   s [] = [1]; s (9:ds) = 1 : s ds; s (d:ds) = 3*d : ds
-- Reinhard Zumkeller, Jan 13 2014

Select[Range[2000], IntegerQ[Log[3, Times @@ (IntegerDigits[#])]] &]

from sympy import integer_log
def A174813(n):
    m = integer_log(k:=(n<<1)+1,3)[0]
    return sum(3**((k-3**m)//(3**j<<1)%3)*10**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A061426 Geometric mean of the digits = 2. In other words, the product of the digits is = 2^k where k is the number of digits.

Original entry on oeis.org

2, 14, 22, 41, 118, 124, 142, 181, 214, 222, 241, 412, 421, 811, 1128, 1144, 1182, 1218, 1224, 1242, 1281, 1414, 1422, 1441, 1812, 1821, 2118, 2124, 2142, 2181, 2214, 2222, 2241, 2412, 2421, 2811, 4114, 4122, 4141, 4212, 4221, 4411, 8112, 8121, 8211

Offset: 1

Amarnath Murthy, May 03 2001

			124 is a term as the geometric mean of digits is (1*2*4) = 8 = 2^3.

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

Cf. A061427-A061430. A069512 gives another version.

Cf. A028846.

a061426 n = a061426_list !! (n-1)
a061426_list = g [1] where
   g ds = if product ds == 2 ^ length ds
          then foldr (\d v -> 10 * v + d) 0 ds : g (s ds) else g (s ds)
   s [] = [1]; s (8:ds) = 1 : s ds; s (d:ds) = 2*d : ds
-- Reinhard Zumkeller, Jan 13 2014

More terms from Erich Friedman, May 08 2001

A061430 Geometric mean of the digits is an integer: k-digit numbers such that the product of the digits is a number of the form m^k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 19, 20, 22, 28, 30, 33, 40, 41, 44, 49, 50, 55, 60, 66, 70, 77, 80, 82, 88, 90, 91, 94, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 118, 120, 124, 130, 139, 140, 142, 150, 160, 170, 180, 181, 188, 190, 193

Offset: 1

Amarnath Murthy, May 03 2001

			694 is a term as (6*9*4)^(1/3) = 6 is an integer.

T. D. Noe, Table of n, a(n) for n=1..5000

Cf. A061426, A061427, A061428.

a061430 n = a061430_list !! (n-1)
a061430_list = filter g [0..] where
   g u = round (fromIntegral p ** (1 / fromIntegral k)) ^ k == p where
         (p, k) = h (1, 0) u
         h (p, l) 0 = (p, l)
         h (p, l) v = h (p * r, l + 1) v' where (v', r) = divMod v 10
-- Reinhard Zumkeller, Jan 13 2014

Select[Range[0,200],IntegerQ[GeometricMean[IntegerDigits[#]]]&] (* Harvey P. Dale, Feb 15 2012 *)

More terms from Naohiro Nomoto, May 11 2001

A061428 Geometric mean of the digits = 4. In other words, the product of the digits is = 4^k where k is the number of digits.

Original entry on oeis.org

4, 28, 44, 82, 188, 248, 284, 428, 444, 482, 818, 824, 842, 881, 1488, 1848, 1884, 2288, 2448, 2484, 2828, 2844, 2882, 4188, 4248, 4284, 4428, 4444, 4482, 4818, 4824, 4842, 4881, 8148, 8184, 8228, 8244, 8282, 8418, 8424, 8442, 8481, 8814, 8822, 8841, 12888

Offset: 1

Amarnath Murthy, May 03 2001

			248 is a term as the geometric mean of digits is (2*4*8) = 64 = 4^3.

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

Cf. A061426, A061427, A061429, A061430.

Cf. A028846, A069518.

a061428 n = a061428_list !! (n-1)
a061428_list = g [1] where
   g ds = if product ds == 4 ^ length ds
          then foldr (\d v -> 10 * v + d) 0 ds : g (s ds) else g (s ds)
   s [] = [1]; s (8:ds) = 1 : s ds; s (d:ds) = 2*d : ds
-- Reinhard Zumkeller, Jan 13 2014

from math import prod
from sympy.utilities.iterables import multiset_combinations, multiset_permutations
def auptod(maxdigits):
  n, digs, alst, powsexps2 = 0, 1, [], [(1, 0), (2, 1), (4, 2), (8, 3)]
  for digs in range(1, maxdigits+1):
    target, okdigs = 4**digs, set()
    mcstr = "".join(str(d)*(digs//max(1, r//2)) for d, r in powsexps2)
    for mc in multiset_combinations(mcstr, digs):
      if prod(map(int, mc)) == target:
        n += 1
        okdigs |= set("".join(mp) for mp in multiset_permutations(mc, digs))
    alst += sorted(map(int, okdigs))
  return alst
print(auptod(4)) # Michael S. Branicky, Apr 28 2021

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

A069516 Geometric mean of digits = 3 and digits are in nondecreasing order.

Original entry on oeis.org

3, 19, 33, 139, 333, 1199, 1339, 3333, 11399, 13339, 33333, 111999, 113399, 133339, 333333, 1113999, 1133399, 1333339, 3333333, 11119999, 11133999, 11333399, 13333339, 33333333, 111139999, 111333999, 113333399, 133333339, 333333333, 1111199999, 1111339999

Offset: 1

Amarnath Murthy, Mar 30 2002

No number is obtainable by permuting the digits of other members - only one with ascending order of digits is included.

			1339 belongs to this sequence but 1933 does not.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A061427, A069512, A069518.

a = {}; b = 3; Do[c = Apply[ Times, IntegerDigits[n]]/b^Floor[ Log[10, n] + 1]; If[c == 1 && Position[a, FromDigits[ Sort[ IntegerDigits[n]]]] == {}, Print[n]; a = Append[a, n]], {n, 1, 10^8}]

from math import prod
from sympy.utilities.iterables import multiset_combinations
def aupton(terms):
  n, digits, alst, powsexps3 = 0, 1, [], [(1, 0), (3, 1), (9, 2)]
  while n < terms:
    target = 3**digits
    mcstr = "".join(str(d)*(digits//max(1, r)) for d, r in powsexps3)
    for mc in multiset_combinations(mcstr, digits):
      if prod(map(int, mc)) == target:
        n += 1
        alst.append(int("".join(mc)))
        if n == terms: break
    else: digits += 1
  return alst
print(aupton(31)) # Michael S. Branicky, Apr 28 2021

Edited and extended by Robert G. Wilson v, Apr 01 2002

Name edited and a(30) and beyond from Michael S. Branicky, Apr 28 2021

A285094 Corresponding values of geometric means of digits of numbers from A061430.

Original entry on oeis.org

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

Offset: 0

Jaroslav Krizek, Apr 14 2017

Jaroslav Krizek, Table of n, a(n) for n = 0..3000

Cf. A061430 (numbers with integer geometric mean of digits in base 10).

Sequences of numbers n such that a(n) = k for k = 0 - 9: A011540 (k = 0), A002275 (k = 1), A061426 (k = 2), A061427 (k = 3), A061428 (k = 4), A002279 (k = 5), A061429 (k = 6), A002281 (k = 7), A002282 (k = 8), A002283 (k = 9).

[0] cat [Floor(&*Intseq(n) ^ (1/#Intseq(n))): n in [1..100000] | IsIntegral(&*Intseq(n) ^ (1/#Intseq(n)))];

cp's OEIS Frontend

A174813 a(n) = number whose product of digits equals a power of 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A061426 Geometric mean of the digits = 2. In other words, the product of the digits is = 2^k where k is the number of digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Extensions

A061430 Geometric mean of the digits is an integer: k-digit numbers such that the product of the digits is a number of the form m^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A061428 Geometric mean of the digits = 4. In other words, the product of the digits is = 4^k where k is the number of digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Python

Extensions

A069516 Geometric mean of digits = 3 and digits are in nondecreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A285094 Corresponding values of geometric means of digits of numbers from A061430.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma