cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Oct 19 2019

Keywords

Comments

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

Examples

			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.

References

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

Links

Crossrefs

Subsequences are: A328544, A328560, A328561.

Programs

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

Formula

A328560 union A328561.

A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.

Original entry on oeis.org

1, 5, 6, 10, 16, 24, 25, 30, 36, 40, 50, 51, 60, 70, 76, 90, 92, 100, 125, 176, 195, 240, 249, 250, 363, 375, 376, 430, 490, 500, 501, 510, 546, 556, 560, 568, 570, 600, 620, 624, 625, 648, 680, 730, 749, 750, 760, 810, 875, 909, 930, 972, 975, 976, 990, 999, 1000, 1001, 1010, 1636, 1680, 1930, 2354, 2400, 2490, 2500, 2510, 2512, 2943, 3000
Offset: 1

Views

Author

Scott R. Shannon, Nov 10 2019

Keywords

Comments

Inspired by A328095. To avoid all primes being in the sequence the divisors of k includes k itself.
Contains 10^k, 5*10^k and 6*10^k for all k, 3*10^k, 4*10^k, 7*10^k and 9*10^k for all odd k. - Robert Israel, Nov 11 2019

Examples

			16 is in the sequence as the divisors of 16 are 1,2,4,8,16, and 16*(1*2*4*8*16) = 16*1024 = 16384, and '16384' contains '16' as a substring.
30 is in the sequence as the divisors of 30 are 1,2,3,5,6,10,15,30, and 30*(1*2*3*5*6*10*15*30) = 30*810000 = 24300000, and '24300000' contains '30' as a substring.

Links

  • Scott R. Shannon, Table of n, a(n) for n = 1..1000. Note when searching for these numbers one needs to use arbitrary precision packages; the product for 24570000 has 1486 digits.

Crossrefs

Cf. A007955, A328095, A326806, A328544.
The sequence of primes contained in their squares is A115738.

Programs

  • Magma
    a:=[]; for k in [1..3000] do t:=IntegerToString(k*(&*Divisors(k))); s:=IntegerToString(k); if s in t then Append(~a,k); end if; end for; a; // Marius A. Burtea, Nov 10 2019
  • Mathematica
    f[n_] := n^(1+DivisorSigma[0, n]/2); aQ[n_] := SequenceCount[IntegerDigits[f[n]], IntegerDigits[n]] > 0; Select[Range[3000], aQ] (* Amiram Eldar, Nov 10 2019 *)

A307751 Numbers k such that the number m multiplied by the product of all its digits contains k as a substring, where m = k multiplied by the product of all its digits.

Original entry on oeis.org

0, 1, 5, 6, 7, 11, 19, 79, 84, 111, 123, 176, 232, 396, 1111, 11111, 111111, 331788, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111
Offset: 1

Views

Author

Scott R. Shannon, Nov 10 2019

Keywords

Comments

Inspired by A328095. Like A328095 this sequence contains all the repunits. These numbers could be called 'Two-step Revenant numbers'. It is unknown if 331788 is the last non-repunit.

Examples

			79 is in the sequence as m = 79*7*9 = 4977, and 4977*4*9*7*7 = 8779428, and '8779428' contains '79' as a substring.
331788 is in the sequence as m = 331788*3*3*1*7*8*8 = 1337769216, and 1337769216*1*3*3*7*7*6*9*2*1*6 = 382291633317888, and '382291633317888' contains '331788' as a substring.

Links

  • Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019.
  • Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019. [Cached copy, pdf file, with permission]

Crossrefs

Cf. A328095, A326806, A328544.

Programs

  • Magma
    a:=[0]; f:=func; for k in [1..1200000] do t:=IntegerToString(f(f(k))); s:=IntegerToString(k); if s in t then Append(~a,k); end if; end for; a; // Marius A. Burtea, Nov 10 2019
  • Mathematica
    f[n_] := n * Times @@ IntegerDigits[n]; aQ[n_] := SequenceCount[IntegerDigits[ f[f[n]] ], IntegerDigits[n]] > 0; Select[Range[0, 10^6], aQ] (* Amiram Eldar, Nov 10 2019 *)

Extensions

a(24)-a(27) from Giovanni Resta, Nov 15 2019
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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