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.

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A061955 Numbers n such that n divides the (left) concatenation of all numbers <= n written in base 2 (most significant digit on right).

Original entry on oeis.org

1, 3, 7, 29, 375, 545, 971, 1643, 37801, 435805, 554423, 565947, 645915, 733533, 871927, 9703985
Offset: 1

Views

Author

Larry Reeves (larryr(AT)acm.org), May 24 2001

Keywords

Comments

This sequence differs from A029519 in that all least significant zeros are kept during concatenation.
No more terms < 10^7. - Lars Blomberg, Aug 31 2011

Examples

			7654321 -> (111)(011)(101)(001)(11)(01)(1) base 2 ->11101110100111011 base 2 = 122171 and 7 divides 122171.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 2; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[Reverse[IntegerDigits[#, b]], c], b], #] &] (* Robert Price, Mar 07 2020 *)
  • PARI
    is(n) = my(t=[]); for(k=1, n, t=concat(Vecrev(binary(k)), t)); if(Mod(subst(Pol(t), x, 2), n)==0, return(1), return(0)) \\ Felix Fröhlich, Jul 06 2017

Extensions

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002; Aug 25 2002
a(13)-a(16) from Lars Blomberg, Aug 31 2011

A029455 Numbers k that divide the (right) concatenation of all numbers <= k written in base 10 (most significant digit on left).

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 12, 15, 18, 20, 25, 27, 30, 36, 45, 50, 54, 60, 69, 75, 90, 100, 108, 120, 125, 135, 150, 162, 180, 200, 216, 225, 248, 250, 270, 300, 324, 360, 375, 405, 450, 470, 500, 540, 558, 600, 648, 675, 710, 750, 810, 900, 1000, 1053, 1080, 1116
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Numbers k such that k divides A007908(k).

Examples

			k = 13 is not a term since 12345678910111213 is not divisible by 13.

Links

Crossrefs

Cf. A007908.
Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978, A332580.
See A171785 for numbers that divide the concatenation of a(1) through a(n).

Programs

  • Mathematica
    b = 10; c = {}; Select[Range[10^5], Divisible[FromDigits[c = Join[c, IntegerDigits[#, b]], b], #] &] (* Robert Price, Mar 11 2020 *)
Select[Range[1200],Divisible[FromDigits[Flatten[IntegerDigits/@Range[#]]],#]&] (* Harvey P. Dale, Dec 31 2020 *)
nxt[{rc_,n_}]:={rc*10^IntegerLength[n+1]+n+1,n+1}; Select[NestList[nxt,{1,1},1200],Mod[#[[1]],#[[2]]]==0&][[;;,2]] (* Harvey P. Dale, Sep 26 2023 *)
  • PARI
    c=0;for(d=1,1e9,for(n=d,-1+d*=10,(c=c*d+n)%n || print1(n","));d--) \\ M. F. Hasler, Sep 11 2011
  • Python
    A029455_list, r = [], 0
for n in range(1,10**4+1):
    r = r*10**len(str(n))+n
    if not (r % n):
        A029455_list.append(n) # Chai Wah Wu, Nov 05 2014
  • Python
    def concat_mod(base, k, mod):
  total, digits, n1 = 0, 1, 1
  while n1 <= k:
    n2, p = min(n1*base-1, k), n1*base
    # Compute ((p-1)*n1+1)*p**(n2-n1+1)-(n2+1)*p+n2 divided by (p-1)**2.
    # Since (a//b)%mod == (a%(b*mod))//b, compute the numerator mod (p-1)**2*mod.
    tmp = pow(p, n2-n1+1, (p-1)**2*mod)
    tmp = ((p-1)*n1+1)*tmp-(n2+1)*p+n2
    tmp = (tmp%((p-1)**2*mod))//(p-1)**2
    total = (total*pow(p, n2-n1+1, mod)+tmp)%mod
    digits, n1 = digits+1, p
  return total
for k in range(1, 10**10+1):
  if concat_mod(10, k, k) == 0: print(k) # Jason Yuen, Jan 27 2024

A029460 Numbers k that divide the (right) concatenation of all numbers <= k written in base 15 (most significant digit on left).

Original entry on oeis.org

1, 3, 5, 7, 15, 20, 21, 25, 28, 33, 35, 36, 45, 49, 60, 63, 68, 75, 84, 100, 105, 140, 147, 175, 180, 196, 204, 207, 225, 245, 252, 287, 300, 315, 375, 420, 441, 500, 525, 540, 588, 675, 700, 724, 735, 756, 805, 875, 879, 900, 945, 980, 1125, 1225, 1260, 1323
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 15; c = {}; Select[Range[10^5], Divisible[FromDigits[c = Join[c, IntegerDigits[#, b]], b], #] &] (* Robert Price, Mar 11 2020 *)

A029466 Numbers k that divide the (right) concatenation of all numbers <= k written in base 21 (most significant digit on left).

Original entry on oeis.org

1, 3, 5, 7, 15, 21, 25, 35, 45, 48, 49, 63, 75, 79, 80, 105, 112, 144, 147, 175, 203, 225, 240, 245, 295, 315, 327, 336, 369, 400, 441, 525, 560, 675, 720, 735, 784, 805, 945, 1008, 1029, 1113, 1200, 1225, 1267, 1323, 1392, 1575, 1680, 1715, 1953, 2043
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 21; c = {}; Select[Range[10^5], Divisible[FromDigits[c = Join[c, IntegerDigits[#, b]], b], #] &] (* Robert Price, Mar 11 2020 *)

A029479 Numbers k that divide the (left) concatenation of all numbers <= k written in base 10 (most significant digit on left).

Original entry on oeis.org

1, 3, 9, 19, 27, 41, 103, 147, 189, 441, 567, 711, 6759, 15353, 24441, 59823, 209903, 1430217, 2848851, 2969973, 13358067, 146247471, 289542573, 1891846557, 2388085659, 4489093899, 5345125899, 5455876131, 9843149241
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

No other terms below 10^10.

Examples

			19 is a term since 19181716151413121110987654321 is divisible by 19.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 10; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[IntegerDigits[#, b], c], b], #] &] (* Robert Price, Mar 12 2020 *)
Select[Range[134*10^5],Divisible[FromDigits[Flatten[IntegerDigits/@Range[#,1,-1]]],#]&] (* Harvey P. Dale, Oct 09 2022 *)
  • Python
    def concat_mod(base, k, mod):
  total, offset, digits, n1 = 0, 0, 1, 1
  while n1 <= k:
    n2, p = min(n1*base-1, k), n1*base
    # Compute ((p-1)*n2-1)*p**(n2-n1+1)-(n1-1)*p+n1 divided by (p-1)**2.
    # Since (a//b)%mod == (a%(b*mod))//b, compute the numerator mod (p-1)**2*mod.
    tmp = pow(p,n2-n1+1,(p-1)**2*mod)
    tmp = ((p-1)*n2-1)*tmp-(n1-1)*p+n1
    tmp = (tmp%((p-1)**2*mod))//(p-1)**2
    total += tmp*pow(base,offset,mod)
    offset, digits, n1 = offset+digits*(n2-n1+1), digits+1, p
  return total%mod
for k in range(1,10**10):
  if concat_mod(10, k, k) == 0: print(k) # Jason Yuen, Jan 14 2024

Extensions

6759 from Andrew Gacek (andrew(AT)dgi.net), Feb 20 2000
More terms from Larry Reeves (larryr(AT)acm.org), May 24 2001
Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
a(18)-a(21) from Max Alekseyev, May 15 2011
a(22)-a(29) from Jason Yuen, Jan 14 2024

A029486 Numbers k that divide the (left) concatenation of all numbers <= k written in base 17 (most significant digit on left).

Original entry on oeis.org

1, 111, 269, 512, 1536, 4499, 4608, 7401, 9216, 23552, 420864, 492544, 3635712, 7943997, 8175681, 11616768, 12344832, 20399616, 49798656, 77196800, 91584000, 162777600, 210598400, 261962240, 305856000, 379123200, 1484144097, 3318331392, 8416131072
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

No other terms below 3*10^10.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029485, A029487-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 17; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[IntegerDigits[#, b], c], b], #] &] (* Robert Price, Mar 12 2020 *)

Extensions

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000
More terms from Larry Reeves (larryr(AT)acm.org), Jan 14 2002
Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
a(13)-a(21) from Max Alekseyev, May 15 2011
a(22)-a(29) from Jason Yuen, Jun 29 2024

A029488 Numbers k that divide the (left) concatenation of all numbers <= k written in base 19 (most significant digit on left).

Original entry on oeis.org

1, 3, 9, 24, 27, 40, 45, 65, 72, 85, 120, 135, 153, 216, 221, 333, 360, 440, 501, 543, 648, 1080, 1240, 1320, 2456, 2493, 3240, 3720, 3960, 4887, 5169, 5427, 6309, 6504, 6981, 7440, 9360, 10320, 11583, 12555, 14391, 16848, 18135, 18255, 22320, 28080, 28183, 28629, 37360, 38313, 51795, 52545, 54405, 59985, 66960, 84240, 89415, 96720, 101520, 118989, 152784, 925520, 3899205, 4756545, 6259240, 9604872, 13659624, 14446344, 17753256, 17784920, 18694341, 19484440, 20518065, 21377495, 23876424, 31787080, 36603495, 60454107, 80052861, 95747177, 98872632
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

a(102) > 3*10^10. - Jason Yuen, Jun 29 2024

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 19; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[IntegerDigits[#, b], c], b], #] &] (* Robert Price, Mar 12 2020 *)

Extensions

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000
More terms from Larry Reeves (larryr(AT)acm.org), May 24 2001
a(47)-a(81) from Max Alekseyev, May 16 2011

A029503 Numbers n such that n divides the (right) concatenation of all numbers <= n written in base 10 (most significant digit on right).

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 101, 107, 153, 167, 277, 414, 486, 858, 1659, 2894, 3093, 4299, 4842, 8838, 22734, 31869, 69492, 361057, 429786, 462018, 859002, 1170801, 1334667, 1663923, 6143512, 6162396, 6212646, 7034661, 8164443
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

This sequence differs from A061939 in that all least significant zeros are kept during concatenation.
Right concatenation, reverse order.
No more terms < 10^7. - Lars Blomberg, Oct 06 2011

Examples

			n = 22 is not a term since 12345678901112131415161718191021222 is not divisible by 22.
See A029495 for other examples.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 10; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[c, Reverse[IntegerDigits[#, b]]], b], #] &] (* Robert Price, Mar 12 2020 *)
  • PARI
    isok(n) = my(s = ""); for (k=1, n, sk = digits(k); forstep (j=#sk, 1, -1, s = concat(s, sk[j]))); (eval(s) % n) == 0; \\ Michel Marcus, Jan 28 2017

Extensions

Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
Additional comments and more terms from Larry Reeves (larryr(AT)acm.org), May 25 2001
a(25)-a(35) from Lars Blomberg, Oct 06 2011

A029512 Numbers n such that n divides the (right) concatenation of all numbers <= n written in base 19 (most significant digit on right).

Original entry on oeis.org

1, 3, 9, 23, 24, 27, 40, 45, 72, 120, 135, 141, 145, 161, 243, 360, 520, 801, 1272, 1279, 1512, 2568, 2681, 3921, 4584, 7155, 8704, 12389, 22405, 27649, 32640, 34720, 40455, 62755, 65568, 89163, 120240, 125671, 144144, 193680, 204080, 241200, 302160, 437040
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

This sequence differs from A061948 in that all least significant zeros are kept during concatenation.

Examples

			See A029495 for example.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 19; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[c, Reverse[IntegerDigits[#, b]]], b], #] &] (* Robert Price, Mar 13 2020 *)

Extensions

Additional comments and more terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2001

A029515 Numbers k such that k divides the (right) concatenation of all numbers <= k written in base 22 (most significant digit on right).

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 11, 14, 18, 21, 35, 75, 97, 102, 105, 175, 196, 291, 343, 375, 455, 485, 1176, 2070, 2151, 3375, 4968, 11375, 17059, 21826, 22326, 28856, 30079, 41265, 47285, 49605, 49966, 67648, 92806, 115074, 143493, 179654, 963874, 1525498, 1527890
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

This sequence differs from A061951 in that all least significant zeros are kept during concatenation.
The next term is > 1460000. - Larry Reeves, Jan 16 2002

Examples

			See A029495 for example.

Links

Crossrefs

Cf. A029447-A029470, A029471-A029494, A029495-A029518, A029519-A029542, A061931-A061954, A061955-A061978.

Programs

  • Mathematica
    b = 22; c = {}; Select[Range[10^4], Divisible[FromDigits[c = Join[c, Reverse[IntegerDigits[#, b]]], b], #] &] (* Robert Price, Mar 13 2020 *)
  • PARI
    lista(nn, m=22) = my(s, t); for(k=1, nn, s=k; while(s, t=t*m+s%m; s\=m); if(t%k==0, print1(k, ", "))); \\ Jinyuan Wang, Dec 05 2020

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2001
Edited and updated by Larry Reeves (larryr(AT)acm.org), Apr 12 2002
Previous Showing 11-20 of 142 results. Next

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