A328560 -id:A328560 - cp's OEIS frontend

A328095 Revenant numbers: numbers k such that k multiplied by the product of all its digits contains k as a substring.

0, 1, 5, 6, 11, 25, 52, 77, 87, 111, 125, 152, 215, 251, 375, 376, 455, 512, 521, 545, 554, 736, 792, 1111, 1125, 1152, 1215, 1251, 1455, 1512, 1521, 1545, 1554, 2115, 2151, 2174, 2255, 2511, 2525, 2552, 4155, 4515, 4551, 5112, 5121, 5145, 5154, 5211, 5225, 5252, 5415, 5451, 5514, 5522, 5541, 5558, 5585, 5855, 8555, 8772, 9375

Offset: 1

N. J. A. Sloane, Oct 19 2019

Sequence is infinite since 11...1 is always a member.

Numbers whose product of digits is a power of ten (and thus necessarily must only have 1,2,4,5,8 as digits) is a subsequence. - Chai Wah Wu, Oct 19 2019

			87 * 8 * 7 = 4872. As the string 87 is visible in the result, 87 is a revenant.
So is 792 because 792 * 7 * 9 * 2 = 99792.
And so is 9375 as 9375 * 9 * 3 * 7 * 5 = 8859375.

Eric Angelini, Posting to Sequence Fans Mailing List, Oct 19 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019.
Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019. [Cached copy, pdf file, with permission]

Subsequences are: A328544, A328560, A328561.

a:= proc(n) option remember; local k; if n=1 then 0 else
      for k from 1+a(n-1) while searchtext(cat(k), cat(k*
      mul(i, i=convert(k, base, 10))))=0 do od: k fi
    end:
seq(a(n), n=1..75);  # Alois P. Heinz, Oct 19 2019

Select[Range[0,10000],SequenceCount[IntegerDigits[#*(Times@@IntegerDigits[ #])],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 19 2019 *)

is_A328095(n)={my(d,m); if(d=vecprod(digits(n))*n, m=10^logint(n, 10)*10; until(n>d\=10,d%m==n && return(1)),!n)} \\ M. F. Hasler, Oct 20 2019

from functools import reduce
from operator import mul
n, A328095_list = 0, []
while len(A328095_list) < 10000:
    sn = str(n)
    if  sn in str(n*reduce(mul,(int(d) for d in sn))):
        A328095_list.append(n)
    n += 1 # Chai Wah Wu, Oct 19 2019

A328560 union A328561.

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 *)

A326833 Numbers whose sum of digits is a power of 10.

Original entry on oeis.org

1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 505, 514, 523, 532, 541, 550, 604, 613

Offset: 1

Alois P. Heinz, Oct 20 2019

Alois P. Heinz, Table of n, a(n) for n = 1..43758 (all terms with up to 9 digits)

Subsequence of A326806.

Cf. A007953, A017173, A028838, A052224, A328560.

q:= n-> (m-> m>0 and m=10^ilog[10](m))(add(i, i=convert(n, base, 10))):
select(q, [$1..1000])[];

isok(n) = my(s=sumdigits(n), k); (s==1) || (s==10) || (ispower(s,,&k) && (k==10)); \\ Michel Marcus, Oct 21 2019

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

A328561 Numbers in A328095 whose product of digits is not a power of 10.

Original entry on oeis.org

0, 5, 6, 77, 87, 375, 376, 736, 792, 2174, 8772, 9375, 11628, 9859155, 23255814, 62227496, 398472522, 3691262781, 6886826188, 517322161894, 774773248793, 2675959368829, 51964667728417, 52446797239186

Offset: 1

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

A subsequence of A328095. All other terms in A328095 have a product of digits that is a power of 10.

a(25) > 10^14. - Giovanni Resta, Oct 27 2019

Cf. A328095, A328560.

a(20)-a(22) from Giovanni Resta, Oct 23 2019

a(23)-a(24) from Giovanni Resta, Oct 24 2019

cp's OEIS Frontend

A328095 Revenant numbers: numbers k such that k multiplied by the product of all its digits contains k as a substring.

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

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A326833 Numbers whose sum of digits is a power of 10.

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

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A328561 Numbers in A328095 whose product of digits is not a power of 10.

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