A028846 -id:A028846 - cp's OEIS frontend

A174781 Duplicate of A028846.

1, 2, 4, 8, 11, 12, 14, 18, 21, 22, 24, 28, 41, 42, 44, 48, 81, 82, 84, 88, 111, 112, 114, 118, 121, 122, 124, 128, 141, 142, 144, 148, 181, 182, 184, 188, 211, 212, 214, 218, 221, 222, 224, 228, 241, 242, 244, 248, 281, 282, 284, 288

Offset: 1

dead

A276037 Numbers using only digits 1 and 5.

Original entry on oeis.org

1, 5, 11, 15, 51, 55, 111, 115, 151, 155, 511, 515, 551, 555, 1111, 1115, 1151, 1155, 1511, 1515, 1551, 1555, 5111, 5115, 5151, 5155, 5511, 5515, 5551, 5555, 11111, 11115, 11151, 11155, 11511, 11515, 11551, 11555, 15111, 15115, 15151, 15155, 15511, 15515

Offset: 1

Vincenzo Librandi, Aug 17 2016

Numbers n such that product of digits of n is a power of 5.

			5551 is in the sequence because all of its digits are 1 or 5 and consequently because the product of digits, 5*5*5*1 = 125 = 5^3 is a power of 5.

Chai Wah Wu, Table of n, a(n) for n = 1..8190

Cf. numbers n such that product of digits of n is a power of k: A028846 (k=2), A174813 (k=3), this sequence (k=5), A276038 (k=6), A276039 (k=7).

Cf. A199985 (a subsequence).

[n: n in [1..20000] | Set(Intseq(n)) subset {1, 5}]; // Vincenzo Librandi, Aug 19 2016

S[0]:= [0]:
for d from 1 to 6 do S[d]:= map(t -> (10*t+1, 10*t+5), S[d-1]) od:
seq(op(S[d]),d=1..6); # Robert Israel, Aug 22 2016

Select[Range[20000], IntegerQ[Log[5, Times@@(IntegerDigits[#])]]&]

a(n) = my(v=[1,5], b=binary(n+1), d=vector(#b-1,i, v[b[i+1]+1])); sum(i=1, #d, d[i] * 10^(#d-i)) \\ David A. Corneth, Aug 22 2016

from itertools import product
A276037_list = [int(''.join(d)) for l in range(1,10) for d in product('15',repeat=l)] # Chai Wah Wu, Aug 18 2016

def A276037(n): return (int(bin(n+1)[3:])<<2)+(10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Jun 28 2025

From Robert Israel, Aug 22 2016: (Start)
a(2n+1) = 10 a(n) + 1.
a(2n+2) = 10 a(n) + 5.
G.f. g(x) satisfies g(x) = 10 (x + x^2) g(x^2) + (x + 5 x^2)/(1 - x^2). (End)

Example changed by David A. Corneth, Aug 22 2016

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

Original entry on oeis.org

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

A028838 Numbers whose sum of digits is a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 10, 11, 13, 17, 20, 22, 26, 31, 35, 40, 44, 53, 62, 71, 79, 80, 88, 97, 100, 101, 103, 107, 110, 112, 116, 121, 125, 130, 134, 143, 152, 161, 169, 170, 178, 187, 196, 200, 202, 206, 211, 215, 220, 224, 233, 242, 251, 259, 260, 268, 277, 286, 295, 301, 305, 310, 314, 323, 332, 341

Offset: 1

N. J. A. Sloane

Marius A. Burtea, Table of n, a(n) for n = 1..10535

Cf. A007953, A028836, A028846.

[1] cat [k: k in [2..341] | PrimeDivisors(&+Intseq(k)) eq [2] or &+Intseq(k) eq 1]; // Marius A. Burtea, Jun 11 2019

Select[Range[400],IntegerQ[Log2[Total[IntegerDigits[#]]]]&] (* Harvey P. Dale, Jul 22 2023 *)

ispp(n) = (n==1) || (isprimepower(n, &p) && (p==2));
isok(n) = ispp(sumdigits(n)); \\ Michel Marcus, Jun 11 2019

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008

A328560 Numbers whose product of digits is a power of 10.

Original entry on oeis.org

1, 11, 25, 52, 111, 125, 152, 215, 251, 455, 512, 521, 545, 554, 1111, 1125, 1152, 1215, 1251, 1455, 1512, 1521, 1545, 1554, 2115, 2151, 2255, 2511, 2525, 2552, 4155, 4515, 4551, 5112, 5121, 5145, 5154, 5211, 5225, 5252, 5415, 5451, 5514, 5522, 5541, 5558, 5585

Offset: 1

Chai Wah Wu, Alois P. Heinz and David A. Corneth, Oct 19 2019

All terms must have only 1, 2, 4, 5, 8 as digits.

A subsequence of A328095.

Alois P. Heinz, Table of n, a(n) for n = 1..27446 (all terms with up to 9 digits)

Cf. A028846, A276037, A284375, A316315, A326833, A328095.

q:= n-> (m-> m>0 and m=10^ilog[10](m))(mul(i, i=convert(n, base, 10))):
select(q, [$1..6000])[];

A284295 Numbers n such that product of digits of n is a power of 9.

Original entry on oeis.org

1, 9, 11, 19, 33, 91, 99, 111, 119, 133, 191, 199, 313, 331, 339, 393, 911, 919, 933, 991, 999, 1111, 1119, 1133, 1191, 1199, 1313, 1331, 1339, 1393, 1911, 1919, 1933, 1991, 1999, 3113, 3131, 3139, 3193, 3311, 3319, 3333, 3391, 3399, 3913, 3931, 3939, 3993

Offset: 1

Jaroslav Krizek, Mar 25 2017

Supersequence of A284294.

			1111 is in the sequence because 1*1*1*1 = 1 = 9^0.

Cf. Numbers n such that product of digits of n is a power of k for k = 0 - 9: A284375 (k = 0), A002275 (k = 1), A028846 (k = 2), A174813 (k = 3), A284323 (k = 4), A276037 (k = 5), A276038 (k = 6), A276039 (k = 7), A284324 (k = 8), this sequence (k = 9).

Set(Sort([n: n in [1..10000], k in [0..20] | &*Intseq(n) eq 9^k]))

FromDigits /@ Select[Join @@ Map[Tuples[{1, 3, 9}, #] &, Range@ 4], IntegerQ@ Log[9, Times @@ #] &] (* Michael De Vlieger, Mar 25 2017 *)

A284324 Numbers k such that product of digits of k is a power of 8.

Original entry on oeis.org

1, 8, 11, 18, 24, 42, 81, 88, 111, 118, 124, 142, 181, 188, 214, 222, 241, 248, 284, 412, 421, 428, 444, 482, 811, 818, 824, 842, 881, 888, 1111, 1118, 1124, 1142, 1181, 1188, 1214, 1222, 1241, 1248, 1284, 1412, 1421, 1428, 1444, 1482, 1811, 1818, 1824, 1842

Offset: 1

Jaroslav Krizek, Mar 26 2017

There are (2 + 4^d)/3 terms with d digits, for each d >= 1. - Robert Israel, Mar 31 2017

			1111 is in the sequence because 1*1*1*1 = 1 = 8^0.

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

Supersequence of A213084.

Cf. Numbers n such that product of digits of n is a power of k for k = 0 - 9: A284375 (k = 0), A002275 (k = 1), A028846 (k = 2), A174813 (k = 3), A284323 (k = 4), A276037 (k = 5), A276038 (k = 6), A276039 (k = 7), this sequence (k = 8), A284295 (k = 9).

Set(Sort([n: n in [1..10000], k in [0..20] | &*Intseq(n) eq 8^k]));

dmax:= 4: # to get all terms with at most dmax digits
B[0,1]:= {1,8}:
B[1,1]:= {2}:
B[2,1]:= {4}:
for d from 2 to dmax do
  for j from 0 to 2 do
    B[j,d]:= map(t -> (10*t+1,10*t+8), B[j,d-1])
        union map(t -> 10*t+4, B[(j+1) mod 3, d-1])
        union map(t->10*t+2, B[(j+2) mod 3, d-1])
od od:
seq(op(sort(convert(B[0,d],list))),d=1..dmax); # Robert Israel, Mar 31 2017

A284375 Numbers whose product of digits is a power of 0.

Original entry on oeis.org

0, 1, 10, 11, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 302, 303

Offset: 1

Jaroslav Krizek, Mar 26 2017

			111 is in the sequence because 1*1*1 = 1 = 0^0.

Union of A011540 and A002275. Supersequence of A007088.

Cf. Numbers n such that product of digits of n is a power of k for k = 0 - 9: this sequence (k = 0), A002275 (k = 1), A028846 (k = 2), A174813 (k = 3), A284323 (k = 4), A276037 (k = 5), A276038 (k = 6), A276039 (k = 7), A284324 (k = 8), A284295 (k = 9), A328560 (k = 10).

Set(Sort([n: n in [1..10000], k in [0..20] | &*Intseq(n) eq 0^k]));

Select[Range[0, 500], Times@@ IntegerDigits[#] <2 &] (* Indranil Ghosh, Mar 26 2017 *)

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

cp's OEIS Frontend

A174781 Duplicate of A028846.

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A276037 Numbers using only digits 1 and 5.

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

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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.

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A028838 Numbers whose sum of digits is a power of 2.

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A328560 Numbers whose product of digits is a power of 10.

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A284295 Numbers n such that product of digits of n is a power of 9.

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A284324 Numbers k such that product of digits of k is a power of 8.

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