A028846 -id:A028846 - cp's OEIS frontend

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

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 *)

A028889 Numbers whose iterated product of digits is a power of 2.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 18, 21, 22, 24, 26, 27, 29, 34, 36, 37, 38, 39, 41, 42, 43, 46, 49, 62, 63, 64, 66, 67, 72, 73, 76, 77, 79, 81, 83, 88, 89, 92, 93, 94, 97, 98, 99, 111, 112, 114, 118, 121, 122, 124, 126, 127, 129, 134, 136, 137, 138, 139, 141, 142, 143, 146, 149, 162, 163, 164, 166, 167

Offset: 1

N. J. A. Sloane

			38 -> 3*8 = 24 -> 2*4 = 8 = 2^3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A031347, A028846, A028836.

p2Q[n_]:=IntegerQ[Log[2,NestWhile[Times@@IntegerDigits[#]&,n,#>9&]]]; Select[ Range[200],p2Q] (* Harvey P. Dale, Sep 19 2019 *)

Extended (and corrected) by Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A272826 Cubes whose digits are powers of 2.

Original entry on oeis.org

1, 8, 21811182184

Offset: 1

Waldemar Puszkarz, May 07 2016

Intersection of A028846 and A000578.
1 and 8, as Fibonacci numbers, are also members of A272827.
There are many squares whose digits are powers of 2: 1,4,81,121,144, to name just a few; there are 102 of them up to 10^12. In contrast, there are very few such cubes, only 3 up to 10^18.
Probably this sequence is finite; further terms have at least 31 digits. - Charles R Greathouse IV, May 19 2016

			21811182184 is a term as its digits are only powers of 2; its cube root is 2794.

Cf. A000578 (cubes), A028846 (numbers whose digits are powers of 2), A272827 (related sequence).

Select[Range[1000000]^3, SubsetQ[{1,2,4,8}, IntegerDigits@#]&]

is(n)=ispower(n,3) && #setintersect(Set(digits(n)),[0,3,5,6,7,9])==0 \\ Charles R Greathouse IV, May 08 2016

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

A381259 Numbers obtained by concatenating powers of 2, sorted into increasing order.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 14, 16, 18, 21, 22, 24, 28, 32, 41, 42, 44, 48, 64, 81, 82, 84, 88, 111, 112, 114, 116, 118, 121, 122, 124, 128, 132, 141, 142, 144, 148, 161, 162, 164, 168, 181, 182, 184, 188, 211, 212, 214, 216, 218, 221, 222, 224, 228, 232, 241, 242, 244, 248, 256, 264

Offset: 1

Stefano Spezia, Feb 18 2025

Take the list {2^i: i >= 0} and concatenate its terms (allowing multiple copies) in any order; then sort the result into increasing order.

The term a(32) = 128 is a power of 2 as well as the concatenation of several powers of 2. - Rémy Sigrist, Feb 20 2025

			11 is a term because it is the concatenation of 1 = 2^0 with itself;
12 is a term because it is the concatenation of 1 = 2^0 with 2 = 2^1;
32 is a term because it is equal to 2^5;
168 is a term because it is the concatenation of 16 = 2^4 with 8 = 2^3.
0 is not a term because it is not a power of 2.

Rémy Sigrist, Table of n, a(n) for n = 1..11921
Rémy Sigrist, PARI program

Supersequence of A028846.
Some subsequences: A000079, A045507, A178664.
Cf. A152242.

PARI
```
\\ See Links section.
```

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.

A272827 Fibonacci numbers whose digits are powers of 2.

Original entry on oeis.org

1, 2, 8, 21, 144, 4181

Offset: 1

Waldemar Puszkarz, May 07 2016

Intersection of A028846 and A000045.
Cubes 1 and 8 are also members of A272826.
a(7), if it exists, is greater than Fibonacci(10^7). - Lars Blomberg, Aug 17 2016

			144 is a term as its digits are only powers of 2 and it is a Fibonacci number (see A000045).

Cf. A000045 (Fibonacci numbers), A028846 (numbers whose digits are powers of 2), A272826 (related sequence).

Select[Fibonacci@Range[2,50000], SubsetQ[{1,2,4,8}, IntegerDigits@#]&]

A272884 Squares whose digits are powers of 2.

Original entry on oeis.org

1, 4, 81, 121, 144, 441, 484, 841, 1444, 8281, 11881, 14884, 28224, 48841, 114244, 128881, 142884, 221841, 228484, 848241, 1121481, 1281424, 1418481, 2184484, 2214144, 8282884, 11142244, 11282881, 18241441, 18818244, 18844281, 21242881, 21818241, 28281124, 82428241, 121242121, 121484484, 124121881

Offset: 1

Waldemar Puszkarz, May 08 2016

Intersection of A000290 and A028846.
Note that in contrast to this sequence, which contains 102 terms up to 10^12, the analogous sequence of cubes (A272826) may contain only 3 in total.
Moreover, the similar sequences for the fourth and fifth perfect powers seem to contain only two terms (1, 81) in the case of the former and only one term (1) in the case of the latter. Higher powers also appear to produce sequences with one (mostly) or two terms only.
Unlike the analogous sequence for cubes, this sequence is heuristically infinite. - Charles R Greathouse IV, May 08 2016
This sequence is infinite because it contains the squares of the numbers of the forms 10*(10^k-1)/3+8 and 100*(10^k-1)/3+59. - Giovanni Resta, May 09 2016
Additionally, this sequence contains the squares of the numbers of the form 1000*(10^k-1)/3 + 809 for k > 2. For k > 2, numbers of the form (1000*(10^k-1)/3 + 809)^2 contains all digits that are powers of 2. - Altug Alkan, May 14 2016

			144 is a term as its digits are only powers of 2 and it is a square, 144 = 12^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000290 (squares), A028846 (numbers whose digits are powers of 2), A272826 (similar sequence for cubes).

Select[Range[12000]^2, SubsetQ[{1, 2, 4, 8}, IntegerDigits@#] &]
Select[Flatten[Table[FromDigits/@Tuples[{1,2,4,8},n],{n,9}]],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Aug 05 2024 *)

is(n)=issquare(n) && #setintersect(Set(digits(n)), [0,3,5,6,7,9])==0 \\ Charles R Greathouse IV, May 08 2016

A383371 Primes having only {1, 2, 4, 8} as digits.

Original entry on oeis.org

2, 11, 41, 181, 211, 241, 281, 421, 811, 821, 881, 1181, 1481, 1811, 2111, 2141, 2221, 2281, 2411, 2441, 4111, 4211, 4241, 4421, 4441, 4481, 8111, 8221, 8821, 11411, 11821, 12211, 12241, 12281, 12421, 12821, 12841, 14221, 14281, 14411, 14821, 18121, 18181, 18211

Offset: 1

Jason Bard, Apr 24 2025

			11 is in this sequence because 1 is an integer power of 2.
13 is not in this sequence because 3 is not an integer power of 2.

Subsequence of A173580. Intersection of A028846 and A000040.
Supersequence of A260267, A260270.
Cf. A381259, A066593.

nmax = 8; Flatten[Table[Select[FromDigits /@ Tuples[{1, 2, 4, 8}, n], PrimeQ], {n, 1, nmax}]]

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

A028889 Numbers whose iterated product of digits is a power of 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A272826 Cubes whose digits are powers of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A381259 Numbers obtained by concatenating powers of 2, sorted into increasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A272827 Fibonacci numbers whose digits are powers of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A272884 Squares whose digits are powers of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI