A276038 -id:A276038 - cp's OEIS frontend

A276037 Numbers using only digits 1 and 5.

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

A284293 Numbers using only digits 1 and 6.

Original entry on oeis.org

1, 6, 11, 16, 61, 66, 111, 116, 161, 166, 611, 616, 661, 666, 1111, 1116, 1161, 1166, 1611, 1616, 1661, 1666, 6111, 6116, 6161, 6166, 6611, 6616, 6661, 6666, 11111, 11116, 11161, 11166, 11611, 11616, 11661, 11666, 16111, 16116, 16161, 16166, 16611, 16616

Offset: 1

Jaroslav Krizek, Mar 25 2017

Product of digits of n is a power of 6; subsequence of A276038.

Prime terms are in A020454.

Alois P. Heinz, Table of n, a(n) for n = 1..16382

Cf. Numbers using only digits 1 and k for k = 0 and k = 2 - 9: A007088 (k = 0), A007931 (k = 2), A032917 (k = 3), A032822 (k = 4) , A276037 (k = 5), this sequence (k = 6), A276039 (k = 7), A213084 (k = 8), A284294 (k = 9).

[n: n in [1..20000] | Set(IntegerToSequence(n, 10)) subset {1, 6}];

Join @@ (FromDigits /@ Tuples[{1,6}, #] & /@ Range[5]) (* Giovanni Resta, Mar 25 2017 *)

def A284293(n): return 5*int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Jun 28 2025

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

A304392 Numbers without a digit 1 with digits in nondecreasing order and the product of digits is a power of 6.

Original entry on oeis.org

6, 23, 49, 66, 229, 236, 334, 389, 469, 666, 2233, 2269, 2349, 2366, 2899, 3338, 3346, 3689, 4499, 4669, 6666, 22239, 22336, 22499, 22669, 23334, 23389, 23469, 23666, 26899, 33368, 33449, 33466, 34899, 36689, 44699, 46669, 66666, 88999, 222299, 222333, 222369, 223349

Offset: 1

David A. Corneth, Jun 20 2018

Applying any of the following to terms in this sequence in any order gives a term from A276038: - Prepend a 1. - Permute digits. - Do nothing.

Subsequence of A276038.

			229 is in the sequence because it has digits in nondecreasing order, no digit 1 and a product of digits 2*2*9 = 36 which is a power of 6.

Michael S. Branicky, Table of n, a(n) for n = 1..10656 (all terms with <= 23 digits)

Cf. A009994, A276038.

Select[Range[10^6], And[FreeQ[#, 1], AllTrue[Differences@ #, # > -1 &], IntegerQ@ Log[6, Times @@ #]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jun 30 2018 *)

is(n) = my(d = digits(n), p = prod(i = 1, #d, d[i])); d[1] >= 2 && vecsort(d) == d && 6^logint(p, 6) == p

from math import prod
from sympy.utilities.iterables import multiset_combinations
def auptod(maxdigs):
  n, digs, alst, targets = 0, 1, [], set(6**i for i in range(1, maxdigs*3))
  for digs in range(1, maxdigs+1):
    mcstr = "".join(str(d)*digs for d in "234689")
    for mc in multiset_combinations(mcstr, digs):
      if prod(map(int, mc)) in targets: alst.append(int("".join(mc)))
  return alst
print(auptod(6)) # Michael S. Branicky, Jun 23 2021

cp's OEIS Frontend

A276037 Numbers using only digits 1 and 5.

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A284293 Numbers using only digits 1 and 6.

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

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

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A304392 Numbers without a digit 1 with digits in nondecreasing order and the product of digits is a power of 6.

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