A038565 -id:A038565 - cp's OEIS frontend

A038564 Numbers whose list of divisors includes each digit 1-9 equally often.

54023, 54203, 55868, 500407, 556744, 769858, 1187666, 1566986, 1875098, 3545924, 5594156, 5733406, 5849014, 5908304, 6100594, 6712006, 7605544, 9106868, 9580654, 10909864, 23456789, 23458679, 23459687, 23465789, 23465987, 23469587, 23475869, 23478569, 23489657

Offset: 1

Naohiro Nomoto

			55868 is a term because, among its divisors (which are 1, 2, 4, 13967, 27934, 55868), each digit from 1 through 9 occurs exactly twice.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A038565.

ld9Q[n_]:=Module[{d=DeleteCases[Sort[Flatten[IntegerDigits/@Divisors[ n]]],0]},Length[ Intersection[ d,Range[ 9]]] == 9&&Length[Union[ Length/@Split[ d]]]==1]; Select[ Range[ 235*10^5],ld9Q] (* Harvey P. Dale, Dec 20 2022 *)

from sympy import divisors
from collections import Counter
def ok(n):
    c = Counter()
    for d in divisors(n, generator=True): c.update(str(d))
    return len(set([c[i] for i in "123456789"])) == 1
print([k for k in range(1, 60000) if ok(k)]) # Michael S. Branicky, Nov 13 2022

More terms from Sascha Kurz, Oct 18 2001

a(26) and beyond from Michael S. Branicky, Nov 13 2022

A045818 Number of times the digits are repeated in A045817.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 8, 2, 2, 2, 8, 2, 4, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 4, 20, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Naohiro Nomoto

e.g. 3602[ 0(2),1(2),2(2),3(2),4(2),5(2),6(2) ], 246506[ 0(2),1(2),2(2),3(2),4(2),5(2),6(2) ], 264533[ 0(2),1(2),2(2),3(2),4(2),5(2),6(2) ],..

N. Nomoto, In the list of divisors of n,...

Cf. A038564, A038565.

More terms from Sean A. Irvine, Sep 26 2011

A045799 In the list of divisors of n (in binary), each digit 0-1 appears equally often.

Original entry on oeis.org

100, 10001, 10100, 11000, 100100, 1000011, 1001001, 1001010, 1001100, 1010010, 1011000, 1100001, 1100100, 1101000, 1110000, 10101010, 11001100, 11011000, 11110000, 100000111, 100001101, 100010101, 100010110, 100011001, 100011100

Offset: 1

Naohiro Nomoto

The corresponding decimal values of the terms are 4, 17, 20, 24, 36, 67, 73, 74, 76, 82, 88, 97, 100, 104, 112, 170, 204, 216, 240, 263, 269, 277, 278, 281, 284, ... - Amiram Eldar, Sep 08 2019

			E.g. divisors of 10100 are (1, 10, 100, 101, 1010, 10100); the numbers of digits (0-1) are [ 0(9),1(9) ].

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. Nomoto, In the list of divisors of n,... [Dead link]

Cf. A038564, A038565, A045810.

fQ[v_] := Length[v] == 2 && v[[1]] == v[[2]]; aQ[n_] := fQ[(Tally @ Flatten @ Join @ IntegerDigits[Divisors[n], 2])[[;; , 2]]]; FromDigits /@ IntegerDigits[Select[ Range[284], aQ], 2] (* Amiram Eldar, Sep 08 2019 *)

A045810 Number of times the digits are repeated in A045799.

Original entry on oeis.org

3, 3, 9, 12, 15, 4, 4, 8, 12, 8, 16, 4, 18, 16, 20, 18, 27, 36, 45, 5, 5, 5, 10, 5, 15, 5, 10, 10, 36, 10, 15, 5, 20, 5, 15, 25, 15, 5, 10, 10, 5, 20, 15, 20, 5, 15, 40, 25, 8, 24, 33, 33, 22, 33, 55, 6, 51, 6, 6, 6, 12, 18, 6, 12, 6, 12, 6, 18, 12, 12, 23, 6, 6, 12, 6, 6, 6, 12, 24

Offset: 1

Naohiro Nomoto

			100[ 0(3),1(3) ], 10001[ 0(3),1(3) ], 10100[ 0(9),1(9) ],....

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. Nomoto, In the list of divisors of n,... [Dead link]

Cf. A038564, A038565, A045799.

f[v_] := If[Length[v] == 2 && v[[1]] == v[[2]], v[[1]], 0]; a[n_] := f[(Tally @ Flatten @ Join @ IntegerDigits[Divisors[n], 2])[[;; , 2]]]; Select[a /@ Range[ 1200], # > 0 &] (* Amiram Eldar, Sep 08 2019 *)

Offset corrected by Amiram Eldar, Sep 08 2019

A045811 In the list of divisors of n (in base 3), each digit 0-2 appears equally often.

Original entry on oeis.org

20, 10022, 10202, 12002, 12020, 12200, 20012, 20102, 20120, 20201, 20210, 20220, 21002, 21200, 22002, 22010, 22020, 22100, 22200, 1210020, 1210022, 1212200, 1220100, 10222011, 12200220, 12202020, 12210210, 20010121, 20010212

Offset: 1

Naohiro Nomoto

			Divisors of 12200 are (1, 10, 100, 122, 1220, 12200); the numbers of digits (0-2) are [0(6), 1(6), 2(6)].

Robert Israel, Table of n, a(n) for n = 1..2500
Naohiro Nomoto, In the list of divisors of n,... [Broken link]

Cf. A038564, A038565, A045812.

f:= proc(n) local D,v,r,i;
  D:= map(op@convert,convert(numtheory:-divisors(n),list),base,3);
  if nops({numboccur(0,D),numboccur(1,D),numboccur(2,D)})=1 then
    r:= convert(n,base,3);
    add(r[i]*10^(i-1),i=1..nops(r))
  fi
end proc:
map(f, [$1..3^8]); # Robert Israel, Nov 14 2020

A045817 Numbers n written in base 7, where in the list of divisors of n (in base 7), each digit 0-6 appears equally often.

Original entry on oeis.org

3602, 246506, 264533, 266405, 303652, 320556, 324255, 325605, 342560, 345064, 345406, 345604, 346340, 362055, 414056, 430462, 434630, 435065, 436430, 436550, 453605, 500426, 500641, 506022, 524360, 524406, 526433, 530632, 532650, 533402

Offset: 1

Naohiro Nomoto

			E.g., divisors of 342560 (base 7) are (1,2,10,20,15463,34256,154630,342560) (all in base 7); the numbers of digits (0-6) are [0(4),1(4),2(4),3(4),4(4),5(4),6(4)].

Robert Israel, Table of n, a(n) for n = 1..161
N. Nomoto, In the list of divisors of n,... [broken link]

Cf. A038564, A038565, A045818.

N:= 7^6:
cv7:= proc(n) local L; L:= convert(n,base,7);
add(L[i]*10^(i-1),i=1..nops(L)) end proc:
V:= Matrix(N,7,datatype=integer[8]):
count:= 0: Res:= NULL:
for i from 1 to N do
  L:= convert(i,base,7);
  M:= Vector[row]([seq(numboccur(d,L),d=0..6)],datatype=integer[8]);
  for r from i to N by i do V[r,..]:= V[r,..] + M od;
  if nops(convert(V[i,..],set))=1 then
    count:= count+1;
    w:= cv7(i);
    Res:= Res,w;
  fi
od:
Res; # Robert Israel, Sep 07 2018

Definition clarified by Robert Israel, Sep 07 2018

A045812 Number of times the digits are repeated in A045811.

Original entry on oeis.org

2, 2, 2, 2, 4, 6, 2, 2, 4, 2, 4, 8, 2, 6, 4, 4, 8, 9, 12, 20, 10, 30, 30, 6, 12, 12, 12, 6, 6, 24, 6, 12, 6, 12, 12, 6, 6, 12, 18, 6, 6, 6, 6, 12, 6, 12, 12, 12, 12, 6, 6, 18, 18, 12, 6, 6, 18, 6, 6, 12, 12, 6, 12, 6, 6, 12, 18, 18, 6, 12, 18, 12, 6, 54, 24, 6, 18, 12, 6, 12, 6, 6, 36, 12, 6

Offset: 0

Naohiro Nomoto

e.g. 20[ 0(2),1(2),2(2) ],10022[ 0(2),1(2),2(2) ],10202[ 0(2),1(2),2(2) ],....

N. Nomoto, In the list of divisors of n,...

Cf. A038564, A038565.

A045813 Base-4 numbers whose list of divisors (in base 4) contains each digit 0-3 the same number of times.

Original entry on oeis.org

320, 20132, 21320, 22033, 23201, 30023, 30203, 30320, 32320, 321202, 1002233, 1002323, 1022033, 1022303, 1032023, 1200323, 1202033, 1202303, 1230203, 1232003, 1300223, 1302023, 1302203, 1320023, 2003201, 2003213, 2003231, 2003312, 2012303, 2013032, 2013212

Offset: 1

Naohiro Nomoto

			Divisors of 32320 are {1, 2, 10, 13, 20, 32, 101, 130, 202, 320, 1010, 1313, 2020, 3232, 13130, 32320} in base 4; each digit appears 12 times.

Naohiro Nomoto, In the list of divisors of n, ...

Cf. A038564, A038565, A045814.

from sympy import divisors
from collections import Counter
from sympy.ntheory import digits
def b4(n): return int("".join(map(str, digits(n, 4)[1:])))
def ok(n):
    c = Counter()
    for d in divisors(n, generator=True): c.update(digits(d, 4)[1:])
    return c[0] == c[1] == c[2] == c[3]
print([b4(k) for k in range(1, 4**7) if ok(k)]) # Michael S. Branicky, Nov 12 2022

A045814 Number of times the digits are repeated in A045813.

Original entry on oeis.org

4, 3, 6, 3, 3, 3, 3, 6, 12, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 6, 2, 2, 2, 2, 4, 6, 4, 2, 2, 6, 8, 2, 4, 4, 6, 2, 2, 4, 6, 2, 4, 16, 2, 2, 4, 4, 4, 2, 8, 2, 2, 4, 2, 12, 2, 2, 12, 10, 2, 4, 2, 4, 12, 2, 6, 12, 10, 4, 4, 4, 4, 2, 4, 4, 4, 4, 8, 4, 8, 4, 4, 2, 2, 2, 6, 4

Offset: 1

Naohiro Nomoto

			Divisors of 320 are {1, 2, 10, 13, 20, 32, 130, 320} in base 4; each digit appears 4 times, so a(1) = 4.
Divisors of 20132 are {1, 2, 10033, 20132} in base 4; each digit appears 3 times, so a(2) = 3.
Divisors of 21320 are {1, 2, 10, 20, 1033, 2132, 10330, 21320} in base 4; each digit appears 6 times, so a(3) = 6

Naohiro Nomoto, In the list of divisors of n,...

Cf. A038564, A038565.

A045815 Integers k such that in the list of divisors of k (in base 6), each digit 0-5 appears equally often.

Original entry on oeis.org

20345, 23405, 30245, 30425, 32045, 40235, 40325, 42035, 43025, 45050, 45450, 50450, 52023, 22043435, 22053335, 23234545, 23344501, 23452345, 24034455, 24243535, 24352435, 24403451, 24433051, 30034454, 30202455, 30334045, 30340454, 30424235

Offset: 1

Naohiro Nomoto

			Divisors of 45050 are (1,2,3,10,4505,13414,22323,45050); the numbers of digits (0-5) are [ 0(4),1(4),2(4),3(4),4(4),5(4) ]

Naohiro Nomoto, In the list of divisors of n,...

Cf. A038564, A038565, A045816.

k := 0:for i from 1 to 35000 do for j from 0 to 5 do a[j] := 0:end do:c := divisors(i):for j from 1 to nops(c) do b := convert(c[j],base,6); for h from 1 to nops(b) do a[ b[h] ] := a[ b[h] ]+1:end do:end do: if(a[0]=a[1] and a[1]=a[2] and a[2]=a[3] and a[4]=a[5]) then k := k+1:q := convert(i,base,6):d[k] := sum(q[o+1]*10^o,o=0..nops(q)-1):end if:end do: q := seq(d[l],l=1..k);
isA045815 := proc(n) local c,j,b,h,a,q ; a := [0,0,0,0,0,0] : c := numtheory[divisors](n): for j from 1 to nops(c) do b := convert(c[j], base, 6); for h from 1 to nops(b) do a[b[h]+1] := a[b[h]+1]+1: end do: end do: if(a[1]=a[2] and a[2]=a[3] and a[3]=a[4] and a[4]=a[5] and a[5]=a[6]) then q := convert(n,base,6) ; add(q[o+1]*10^o,o=0..nops(q)-1) ; else -1 ; end if: end: n := 1: while true do a := isA045815(n) : if a >= 0 then printf("%d, ",a) ; fi ; n := n+1 : od : # R. J. Mathar, Jun 26 2007

More terms from Sascha Kurz, Mar 24 2002
Corrected by R. J. Mathar, Jun 26 2007
More terms from Sean A. Irvine, Sep 26 2011

cp's OEIS Frontend

A038564 Numbers whose list of divisors includes each digit 1-9 equally often.

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A045818 Number of times the digits are repeated in A045817.

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A045799 In the list of divisors of n (in binary), each digit 0-1 appears equally often.

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Mathematica

A045810 Number of times the digits are repeated in A045799.

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A045811 In the list of divisors of n (in base 3), each digit 0-2 appears equally often.

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Maple

A045817 Numbers n written in base 7, where in the list of divisors of n (in base 7), each digit 0-6 appears equally often.

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Programs

Maple

Extensions

A045812 Number of times the digits are repeated in A045811.

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A045813 Base-4 numbers whose list of divisors (in base 4) contains each digit 0-3 the same number of times.

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Python

A045814 Number of times the digits are repeated in A045813.

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