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 1-9 of 9 results.

A088392 a(n)=A069522(n)/n.

Original entry on oeis.org

3, 6, 1, 3, 6, 2, 3, 15, 0, 3, 0, 1, 777, 15, 2, 75, 6, 0, 579, 6, 1, 0, 87, 5, 12, 3885, 0, 75, 69, 1, 322581, 375, 0, 3, 6, 0, 3, 2895, 259, 3, 0, 5, 2327907, 0, 0, 435, 213, 25, 22449, 6, 2, 19425, 1887, 0, 0, 375, 193, 345, 339, 2, 1641, 1612905, 0, 1875, 1554, 0, 3, 15
Offset: 1

Views

Author

Ray Chandler, Sep 28 2003

Keywords

Crossrefs

Cf. A069522.

A069530 Smallest multiple of n with digit sum = 11, or 0 if no such number exists.

Original entry on oeis.org

29, 38, 0, 56, 65, 0, 56, 56, 0, 290, 209, 0, 65, 56, 0, 128, 119, 0, 38, 380, 0, 308, 92, 0, 425, 182, 0, 56, 29, 0, 155, 128, 0, 272, 245, 0, 74, 38, 0, 560, 164, 0, 344, 308, 0, 92, 47, 0, 245, 650, 0, 416, 371, 0, 605, 56, 0, 290, 236, 0, 1037, 434, 0, 128, 65, 0, 335
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

a(n) = 0 if n is divisible by 3, 1111, 1507, 2849, 3367, 4849, 5291 or 7373. - Robert Israel, Feb 14 2024

Links

Crossrefs

Cf. A166311, A069521, A069522, A069523, A069524, A069525, A069526, A069527, A069528, A069529.

Programs

  • Maple
    A069530 := proc(n)
    local m ;
    if modp(n,3) = 0 then
         0 ;
    else
        for m from 1 do
            if digsum(m*n) = 11 then
                return m*n ;
            end if;
        end do:
    end if;
end proc:
seq(A069530(n),n=1..70) ; # R. J. Mathar, Aug 06 2019
  • PARI
    sod(n) = {digs = digits(n); return (sum(i=1, #digs, digs[i]));}
a(n) = {if (n % 3 == 0, return (0)); k = 1; while (sod(k*n) != 11, k++); k;} \\ Michel Marcus, Sep 14 2013

Formula

a(3k) = 0 for k = 1, 2, 3, ....
a(n) = n*A088400(n). - R. J. Mathar, Aug 06 2019

Extensions

More terms from Sascha Kurz, Apr 08 2002

A069523 Smallest multiple of n with digit sum = 4, or 0 if no such number exists, e.g. a(3k)= 0.

Original entry on oeis.org

4, 4, 0, 4, 40, 0, 112, 40, 0, 40, 22, 0, 13, 112, 0, 112, 1003, 0, 1102, 40, 0, 22, 1012, 0, 400, 130, 0, 112, 1102, 0, 31, 1120, 0, 10030, 1120, 0, 0, 1102, 0, 40, 0, 0, 301, 220, 0, 1012, 300001, 0, 120001, 400, 0, 1300, 1001011, 0, 220, 112, 0, 1102, 1003, 0
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

a(3k)=a(37k)=a(41k)=0. - Ray Chandler, Jul 30 2003

Crossrefs

Cf. A052218, A069521, A069522.

Formula

a(n) = n*A088393(n). - R. J. Mathar, Aug 06 2019

Extensions

More terms from Ray Chandler, Jul 30 2003

A069524 Smallest multiple of n with digit sum = 5, or 0 if no such number exists, e.g. a(3k) = a(11k) = 0.

Original entry on oeis.org

5, 14, 0, 32, 5, 0, 14, 32, 0, 50, 0, 0, 104, 14, 0, 32, 221, 0, 1121, 140, 0, 0, 23, 0, 50, 104, 0, 140, 203, 0, 2201, 32, 0, 2210, 140, 0, 0, 3002, 0, 320, 41, 0, 2021, 0, 0, 230, 2021, 0, 22001, 50, 0, 104, 212, 0, 0, 1400, 0, 2030, 1121, 0, 122, 22010, 0, 320, 1040
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

a(3k)= a(11k) =a(37k)=0. - Ray Chandler, Jul 30 2003

Links

Crossrefs

Cf. A052219, A069521, A069522, A069523.

Programs

  • Maple
    nextL:= proc(L)
 local m,q,Lp;
 for m from 1 do
   if L[m] > 0 then
    if m = LinearAlgebra:-Dimension(L) then return <4,0$(m-1),1>
    else Lp:= L;
       Lp[1]:= L[m]-1;
       Lp[2..m]:= 0;
       Lp[m+1]:= L[m+1]+1;
       return Lp;
    fi
   fi
 od;
end proc:
N:= 300:
Agenda:= remove(t -> ormap(s -> (t mod s = 0), [3,11,37,101,137,239]), {$1..N}):
L:= <5>: x:= 5:
A:= Vector(N):
while Agenda <> {} and x < 10^40 do
  x:= add(L[i]*10^(i-1),i=1..LinearAlgebra:-Dimension(L));
  found,Agenda:= selectremove(t -> x mod t = 0, Agenda);
  if found <> {} then
    A[convert(found,list)]:= x;
  fi;
  L:= nextL(L);
od:
convert(A,list): # Robert Israel, Sep 05 2019
  • Mathematica
    Table[If[MemberQ[Table[Mod[n,q],{q,{3,11,37,101,137,239}}],0],0,With[{k=5000},SelectFirst[n*Range[k],Total[IntegerDigits[#]] == 5&]]],{n,70}] (* Harvey P. Dale, Nov 03 2024 *)

Formula

a(n) = n*A088394(n). - R. J. Mathar, Aug 06 2019

Extensions

More terms from Ray Chandler, Jul 30 2003

A069525 Smallest multiple of n with digit sum = 6, or 0 if no such number exists, e.g. a(9k)= 0.

Original entry on oeis.org

6, 6, 6, 24, 15, 6, 42, 24, 0, 60, 33, 24, 312, 42, 15, 240, 51, 0, 114, 60, 42, 132, 1104, 24, 150, 312, 0, 420, 1131, 60, 1023, 2112, 33, 204, 105, 0, 222, 114, 312, 240, 123, 42, 1032, 132, 0, 1104, 141, 240, 12201, 150, 51, 312, 1113, 0, 330, 4200, 114, 4002, 2301
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

In addition to those divisible by 9, all numbers n divisible by 239, 271 or 803 have a(n)=0. - Robert Israel, Sep 04 2019

Links

Crossrefs

Cf. A062220, A069521, A069522, A069523, A069524.

Programs

  • Maple
    N:= 1000: # to get a(1)..a(N)
nextL:= proc(L)
local m,q,Lp;
for m from 1 do
   if L[m] > 0 then
    if m = LinearAlgebra:-Dimension(L) then return <5,0$(m-1),1>
    else Lp:= L;
       Lp[1]:= L[m]-1;
       Lp[2..m]:= 0;
       Lp[m+1]:= L[m+1]+1;
       return Lp;
    fi
   fi
od;
end proc:
nogo:= proc(n) local m,a2,a5,S,S2,S3,i,j;
  a2:= padic:-ordp(n,2);
  a5:= padic:-ordp(n,5);
  m:= numtheory:-order(10,n/(2^a2*5^a5))+max(a2,a5);
  S:= {seq(10^i mod n, i=0..m-1)};
  S2:= {seq(seq(S[i]+S[j] mod n,j=1..i),i=1..nops(S))};
  S3:= {seq(seq(S[i]+ S2[j] mod n, j=1..nops(S2)),i=1..nops(S))};
  evalb(S3 intersect map(t -> -t mod n, S3) = {});
end proc:
Agenda:= remove(t -> (t mod 9=0 or t mod 239=0 or t mod 271=0 or t mod 803=0, {$1..N}):
L:= <6>: x:= 6:
A:= Vector(N):
while Agenda <> {} and x < 10^20 do
  x:= add(L[i]*10^(i-1),i=1..LinearAlgebra:-Dimension(L));
  found,Agenda:= selectremove(t -> x mod t = 0, Agenda);
  if found <> {} then
    A[convert(found,list)]:= x;
  fi;
  L:= nextL(L);
od:
Agenda:= remove(nogo,Agenda);
if Agenda <> {} then printf("Values not found for %a\n",Agenda) fi;
convert(A,list); # Robert Israel, Sep 04 2019
  • Mathematica
    Array[If[AnyTrue[Mod[#, {9, 239, 271, 803}], # == 0 &], 0, Block[{k = 1}, While[Total@ IntegerDigits[k #] != 6, k++]; k #]] &, 59] (* Michael De Vlieger, Sep 04 2019 *)

Formula

a(n) = n*A088395(n). - R. J. Mathar, Aug 06 2019

Extensions

More terms from Ray Chandler, Jul 30 2003

A069526 Smallest multiple of n with digit sum = 7, or 0 if no such number exists, e.g. a(3k)= a(11k) = 0.

Original entry on oeis.org

7, 16, 0, 16, 25, 0, 7, 16, 0, 70, 0, 0, 52, 70, 0, 16, 34, 0, 133, 160, 0, 0, 115, 0, 25, 52, 0, 700, 232, 0, 124, 160, 0, 34, 70, 0, 0, 304, 0, 160, 205, 0, 43, 0, 0, 322, 1222, 0, 2401, 250, 0, 52, 106, 0, 0, 7000, 0, 232, 4012, 0, 61, 124, 0, 1024, 520, 0, 11122, 340
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

Either no multiples of 37 have digit sum 7 or their first values are > 10000000. - Larry Reeves (larryr(AT)acm.org), Jul 02 2002
No multiples of 3, 11, 37, 101 or 271 have digit sum 7. - Robert Israel, Feb 13 2024

Links

Crossrefs

Cf. A052221, A069521, A069522, A069523, A069524, A069525, A069527, A069528.

Programs

  • Maple
    unfinished:= true: V:= Vector(300): V0:= select(t -> igcd(t, 3*11*37*101*271) = 1, {$1..300}):
for i1 from 0 while unfinished do
  for i2 from 0 to i1 while unfinished do
    for i3 from 0 to i2 while unfinished do
      for i4 from 0 to i3 while unfinished do
        for i5 from 0 to i4 while unfinished do
          for i6 from 0 to i5 while unfinished do
            for i7 from 0 to i6 while unfinished do
              v:= 10^i1 + 10^i2 + 10^i3 + 10^i4 + 10^i5 + 10^i6 + 10^i7;
              dv:= numtheory:-divisors(v);
              for s in V0 intersect dv do
                V[s]:= v;
              od;
              V0:= V0 minus dv;
              unfinished:= evalb(V0 <> {});
od od od od od od od:
convert(V,list); # Robert Israel, Feb 13 2024

Formula

a(n)= n*A088396(n). - R. J. Mathar, Aug 06 2019

Extensions

More terms from Larry Reeves, Jul 02 2002

A069527 Smallest multiple of n with digit sum = 8, or 0 if no such number exists, e.g. a(3k)= 0.

Original entry on oeis.org

8, 8, 0, 8, 35, 0, 35, 8, 0, 80, 44, 0, 26, 224, 0, 80, 17, 0, 152, 80, 0, 44, 161, 0, 125, 26, 0, 224, 116, 0, 62, 224, 0, 170, 35, 0, 0, 152, 0, 80, 1025, 0, 215, 44, 0, 1610, 611, 0, 1421, 350, 0, 260, 53, 0, 440, 224, 0, 116, 413, 0, 305, 62, 0, 512, 260, 0, 134, 1700, 0
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

The number ABCDEF (A through F are digits) is divisible by 37 if the number XYZ (where X=A+D, Y=B+E, Z=C+F) is divisible by 37. If the digit sum of XYZ is S, then the digit sum of ABCDEF is S+9k for some k. A quick check of all multiples of 37 with three or fewer digits shows that none have a digit sum of 8. Thus no multiple of 37 has a digit sum of 8 and a(37) is undefined as is a(37p) for all p. - Christopher Lund (clund(AT)san.rr.com), Apr 16 2002
a(n) = 0 if n is a multiple of 3, 37, 271 or 4649. - Robert Israel, Feb 14 2024

Links

Crossrefs

Cf. A069521, A069522, A069523, A069524, A069525, A069526.

Programs

  • Maple
    unfinished:= true: V:= Vector(1000): V0:= select(t -> igcd(t, 3*37*271*4649) = 1, {$1..1000}):
for i1 from 0 while unfinished do
  for i2 from 0 to i1 while unfinished do
    for i3 from 0 to i2 while unfinished do
      for i4 from 0 to i3 while unfinished do
        for i5 from 0 to i4 while unfinished do
          for i6 from 0 to i5 while unfinished do
            for i7 from 0 to i6 while unfinished do
              for i8 from 0 to i7 while unfinished do
              v:= 10^i1 + 10^i2 + 10^i3 + 10^i4 + 10^i5 + 10^i6 + 10^i7 + 10^i8;
              dv:= numtheory:-divisors(v);
              for s in V0 intersect dv do
                V[s]:= v;
              od;
              V0:= V0 minus dv;
              unfinished:= evalb(V0 <> {});
od od od od od od od od:
convert(V,list); # Robert Israel, Feb 14 2024

Formula

a(n) = n*A088397(n).

Extensions

More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002

A069528 Smallest multiple of n with digit sum = 9, or 0 if no such number exists (e.g., a(11k) = 0).

Original entry on oeis.org

9, 18, 9, 36, 45, 18, 63, 72, 9, 90, 0, 36, 117, 126, 45, 144, 153, 18, 171, 180, 63, 0, 207, 72, 225, 234, 27, 252, 261, 90, 1116, 1152, 0, 306, 315, 36, 333, 342, 117, 360, 1107, 126, 1161, 0, 45, 414, 423, 144, 441, 450, 153, 1404, 1431, 54, 0, 504, 171, 522
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

a(n) = 0 if n is a multiple of 11, 101, 271, 999 or 4649. - Robert Israel, Feb 14 2024

Links

Crossrefs

Cf. A069521, A069522, A069523, A069524, A069525, A069526, A069527.

Programs

  • Maple
    unfinished:= true: V:= Vector(1000): V0:= select(t -> igcd(t, 11*101*271*4649) = 1 and t mod 999 <> 0, {$1..1000}):
for i1 from 0 while unfinished do
  for i2 from 0 to i1 while unfinished do
    for i3 from 0 to i2 while unfinished do
      for i4 from 0 to i3 while unfinished do
        for i5 from 0 to i4 while unfinished do
          for i6 from 0 to i5 while unfinished do
            for i7 from 0 to i6 while unfinished do
            for i8 from 0 to i7 while unfinished do
            for i9 from 0 to i8 while unfinished do
              v:= 10^i1 + 10^i2 + 10^i3 + 10^i4 + 10^i5 + 10^i6 + 10^i7 + 10^i8 + 10^i9;
              dv:= numtheory:-divisors(v);
              for s in V0 intersect dv do
                V[s]:= v;
              od;
              V0:= V0 minus dv;
              unfinished:= evalb(V0 <> {});
od od od od od od od od od:
convert(V,list); # Robert Israel, Feb 14 2024

Extensions

More terms from Sascha Kurz, Apr 08 2002

A069529 Smallest multiple of n with digit sum = 10, or 0 if no such number exists, e.g. a(3k)= 0.

Original entry on oeis.org

19, 28, 0, 28, 55, 0, 28, 64, 0, 190, 55, 0, 91, 28, 0, 64, 136, 0, 19, 280, 0, 154, 46, 0, 325, 208, 0, 28, 145, 0, 217, 64, 0, 136, 280, 0, 37, 190, 0, 280, 82, 0, 172, 352, 0, 46, 235, 0, 343, 550, 0, 208, 424, 0, 55, 280, 0, 406, 118, 0, 244, 1054, 0, 64, 325, 0, 1072
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2002

Keywords

Comments

a(n) = 0 if n is a multiple of 3, 1111, 2849, 3367, 4649 or 5291.

Links

Crossrefs

Cf. A069521, A069522, A069523, A069524, A069525, A069526, A069527, A069528.

Programs

  • Maple
    unfinished:= true: V:= Vector(1000):
V0:= select(t -> igcd(t, 3*4649) = 1 and t mod 1111 <> 0 and t mod 2849 <> 0 and t mod 3367 <> 0 and t mod 5291 <> 0, {$1..1000}):
for i1 from 0 while unfinished do
  for i2 from 0 to i1 while unfinished do
    for i3 from 0 to i2 while unfinished do
      for i4 from 0 to i3 while unfinished do
        for i5 from 0 to i4 while unfinished do
          for i6 from 0 to i5 while unfinished do
            for i7 from 0 to i6 while unfinished do
            for i8 from 0 to i7 while unfinished do
            for i9 from 0 to i8 while unfinished do
            for i10 from 0 to i9 while unfinished do
              v:= 10^i1 + 10^i2 + 10^i3 + 10^i4 + 10^i5 + 10^i6 + 10^i7 + 10^i8 + 10^i9 + 10^i10;
              if convert(convert(v,base,10),`+`) <> 10 then next fi;
              dv:= numtheory:-divisors(v);
              for s in V0 intersect dv do
                V[s]:= v;
              od;
              V0:= V0 minus dv;
              unfinished:= evalb(V0 <> {});
od od od od od od od od od od:
convert(V,list); # Robert Israel, Feb 14 2024

Extensions

More terms from Sascha Kurz, Apr 08 2002
Showing 1-9 of 9 results.

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