A174813 -id:A174813 - cp's OEIS frontend

A028846 Numbers whose product of digits is a power of 2.

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, 411, 412, 414, 418, 421, 422, 424, 428, 441, 442, 444, 448

Offset: 1

N. J. A. Sloane

Numbers using only digits 1, 2, 4, and 8. - Michel Lagneau, Dec 01 2010

			28 is in the sequence because 2*8 = 2^4. - _Michel Lagneau_, Dec 01 2010

Bo Gyu Jeong, Table of n, a(n) for n = 1..5000

Cf. A007954, A028889, A028838.

Cf. A174813, A061426.

a028846 n = a028846_list !! (n-1)
a028846_list = f [1] where
   f ds = foldr (\d v -> 10 * v + d) 0 ds : f (s ds)
   s [] = [1]; s (8:ds) = 1 : s ds; s (d:ds) = 2*d : ds
-- Reinhard Zumkeller, Jan 13 2014

Select[Range[1000], IntegerQ[Log[2, Times @@ (IntegerDigits[#])]] &] (* Michel Lagneau, Dec 01 2010 *)

is(n)=#setminus(Set(digits(n)), [1,2,4,8])==0 \\ Charles R Greathouse IV, Apr 24 2025

from itertools import count, islice, product
def agen(): yield from (int("".join(p)) for d in count(1) for p in product("1248", repeat=d))
print(list(islice(agen(), 64))) # Michael S. Branicky, Aug 21 2022

def A028846(n):
    m = (k:=3*n+1).bit_length()-1>>1
    return sum(10**j<<((k-(1<<(m<<1)))//(3<<(j<<1))&3) for j in range(m)) # Chai Wah Wu, Jun 28 2025

Given a(0) = 0 and n = 4k - r, where 0 <= r <= 3, a(n) = 10*a(k-1) + 2^(3-r). - Clinton H. Dan, Aug 21 2022

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

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

A329761 Primes having only {1, 3, 9} as digits.

Original entry on oeis.org

3, 11, 13, 19, 31, 113, 131, 139, 191, 193, 199, 311, 313, 331, 911, 919, 991, 1193, 1319, 1399, 1913, 1931, 1933, 1993, 1999, 3119, 3191, 3313, 3319, 3331, 3391, 3911, 3919, 3931, 9133, 9199, 9311, 9319, 9391, 9931, 11113, 11119, 11131, 11311, 11393, 11399

Offset: 1

Alois P. Heinz, Nov 20 2019

Original name was: Primes whose product of decimal digits is a power of 3.

Primes whose digit set is a subset of {1,3,9}.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Marianne Freiberger, Primes without 7s.
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Subsequence of A030096.

Cf. A000040, A000244, A007954, A020457, A174813.

[p: p in PrimesUpTo(12000) | Set(Intseq(p)) subset [1,3,9]]; // Vincenzo Librandi, Jan 02 2019

Select[Prime[Range[1500]],IntegerQ[Log[3,Times@@IntegerDigits[#]]]&] (* or *) Table[Select[FromDigits/@Tuples[{1,3,9},n],PrimeQ],{n,5}]// Flatten (* Harvey P. Dale, Dec 31 2019 *)

{ A000040 } intersect { A174813 }.

a(n) in { A000040 } and A007954(a(n)) in { A000244 }.

Name changed by Sean A. Irvine, Jul 20 2025

A061427 Geometric mean of the digits = 3. In other words, the product of the digits is = 3^k where k is the number of digits.

Original entry on oeis.org

3, 19, 33, 91, 139, 193, 319, 333, 391, 913, 931, 1199, 1339, 1393, 1919, 1933, 1991, 3139, 3193, 3319, 3333, 3391, 3913, 3931, 9119, 9133, 9191, 9313, 9331, 9911, 11399, 11939, 11993, 13199, 13339, 13393, 13919, 13933, 13991, 19139, 19193

Offset: 1

Amarnath Murthy, May 03 2001

			319 is a term as the geometric mean of digits is (3*1*9) = 27 = 3^3.

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

Cf. A061426-A061430.

Cf. A174813.

a061427 n = a061427_list !! (n-1)
a061427_list = g [1] where
   g ds = if product ds == 3 ^ length ds
          then foldr (\d v -> 10 * v + d) 0 ds : g (s ds) else g (s ds)
   s [] = [1]; s (9:ds) = 1 : s ds; s (d:ds) = 3*d : ds
-- Reinhard Zumkeller, Jan 13 2014

Select[Range[20000],GeometricMean[IntegerDigits[#]]==3&] (* Harvey P. Dale, Dec 11 2011 *)

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

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 *)

A284323 Numbers k such that product of digits of k is a power of 4.

Original entry on oeis.org

1, 4, 11, 14, 22, 28, 41, 44, 82, 88, 111, 114, 122, 128, 141, 144, 182, 188, 212, 218, 221, 224, 242, 248, 281, 284, 411, 414, 422, 428, 441, 444, 482, 488, 812, 818, 821, 824, 842, 848, 881, 884, 1111, 1114, 1122, 1128, 1141, 1144, 1182, 1188, 1212, 1218

Offset: 0

Jaroslav Krizek, Mar 25 2017

			1111 is in the sequence because 1*1*1*1 = 1 = 4^0.

Supersequence of A032822.

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), this sequence (k = 4), A276037 (k = 5), A276038 (k = 6), A276039 (k = 7), A284324 (k = 8), A284295 (k = 9).

Set(Sort([n: n in [1..10000], k in [0..20] | &*Intseq(n) eq 4^k]));

FromDigits /@ Select[Join @@ Map[Tuples[2^Range[0, 3], #] &, Range@ 4], IntegerQ@ Log[4, Times @@ #] &] (* Michael De Vlieger, Mar 25 2017 *)

A316315 Numbers k such that the product of digits of k is a power of 12.

Original entry on oeis.org

1, 11, 26, 34, 43, 62, 111, 126, 134, 143, 162, 216, 223, 232, 261, 289, 298, 314, 322, 341, 368, 386, 413, 431, 449, 466, 494, 612, 621, 638, 646, 664, 683, 829, 836, 863, 892, 928, 944, 982, 1111, 1126, 1134, 1143, 1162, 1216, 1223, 1232, 1261, 1289, 1298

Offset: 1

Isaac Weiss and Henry Potts-Rubin, Jun 29 2018

			466 is in the sequence because 4*6*6 = 144 = 12^2.

Alois P. Heinz, Table of n, a(n) for n = 1..64235

Cf. A284375, A002275, A028846, A174813, A284323, A276037, A276038, A276039, A284324, A284295, A328560.

FromDigits /@ Select[Join @@ Map[Tuples[{1, 2, 3, 4, 6, 8, 9}, #] &, Range@4], IntegerQ@Log[12, Times @@ #] &]

Two duplicate terms removed by Alois P. Heinz, Oct 20 2019

A385324 Numbers whose digits are all powers of the same single-digit base.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 28, 31, 33, 39, 41, 42, 44, 48, 51, 55, 61, 66, 71, 77, 81, 82, 84, 88, 91, 93, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 124, 128, 131, 133, 139, 141, 142, 144, 148

Offset: 1

Stefano Spezia, Jun 25 2025

			84 is a term since its digits 8 and 4 are both powers of 2.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A028846, A174813, A276037, A276039, A284293.

Cf. A385351 (subsequence)

Select[Range[0,148],SubsetQ[{0},dig=IntegerDigits[#]]||SubsetQ[{1,2,4,8},dig]||SubsetQ[{1,3,9},dig]||SubsetQ[{1,5},dig]||SubsetQ[{1,6},dig]||SubsetQ[{1,7},dig] &]

{0} U A028846 U A174813 U A276037 U A276039 U A284293.

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A028846 Numbers whose product of digits is a power of 2.

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A276037 Numbers using only digits 1 and 5.

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A329761 Primes having only {1, 3, 9} as digits.

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A061427 Geometric mean of the digits = 3. In other words, the product of the digits is = 3^k where k is the number of digits.

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

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