A217398 -id:A217398 - cp's OEIS frontend

A131835 Numbers starting with 1.

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142

Offset: 1

Andrew Good (yipes_stripes(AT)yahoo.com), Jul 20 2007

The lower and upper asymptotic densities of this sequence are 1/9 and 5/9, respectively. - Amiram Eldar, Feb 27 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz and Michael Someck, Four quotient set gems, The American Mathematical Monthly, Vol. 121, No. 7 (2014), pp. 590-598; arXiv preprint, arXiv:1312.1036 [math.NT], 2013.
Index entries for 10-automatic sequences.

Subsequence of A011531.
Disjoint union of A045707 and A206286.
Cf. A000030, A000027, A002275, A262390 (permutation).
Cf. A217394, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

a131835 n = a131835_list !! (n-1)
a131835_list = concat $
               iterate (concatMap (\x -> map (+ 10 * x) [0..9])) [1]
-- Reinhard Zumkeller, Jul 16 2014

isA131835 := proc(n) if op(-1,convert(n,base,10)) = 1 then true; else false ; fi ; end: for n from 1 to 300 do if isA131835(n) then printf("%d, ",n) ; fi ; od : # R. J. Mathar, Jul 24 2007

Select[Range[150], IntegerDigits[#][[1]] == 1 &] (* Amiram Eldar, Feb 27 2021 *)

a(n, {base=10}) = my (o=1); while (n>o, n-=o; o*=base); return (o+n-1) \\ Rémy Sigrist, Jun 23 2017

a(n) = n--; s = #digits(9*n+1); n + 8 * (10^(s-1))/9 + 1/9 \\ David A. Corneth, Jun 23 2017

nxt(n) = my(d = digits(n+1)); if(d[1]==1, n+1, 10^#d) \\ David A. Corneth, Jun 23 2017

def A131835(n): return n+(10**(len(str(9*n-8))-1)<<3)//9 # Chai Wah Wu, Dec 07 2024

A000030(a(n)) = 1. - Reinhard Zumkeller, Jul 16 2014
a(A002275(n)+1) = 10^n for any n >= 0. - Rémy Sigrist, Jun 23 2017
a(n) = n + (8*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 16 2023

More terms from R. J. Mathar, Jul 24 2007

A217394 Numbers starting with 2.

Original entry on oeis.org

2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/18 and 10/27, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A006092, A011532.
Subsequences include: A045708, A045726, A077327, A077678, A106412, A106422.
Cf. A131835, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

Select[Range[300], IntegerDigits[#][[1]] == 2 &] (* T. D. Noe, Oct 02 2012 *)

def agen():
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(10**digits): yield adder + i
    digits, adder = digits+1, adder*10
g = agen()
print([next(g) for i in range(54)]) # Michael S. Branicky, Feb 20 2021

def A217394(n): return n+(17*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (17*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 15 2023

A217399 Numbers starting with 6.

Original entry on oeis.org

6, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/54 and 10/63, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000870, A011536.
Subsequences include: A045712, A077331, A077682, A106416, A106426.
Cf. A131835, A217394, A217395, A217397, A217398, A217400, A217401, A217402.

Flat(List([0..2],n->List([0..10^n-1],k->6*10^n+k))); # Muniru A Asiru, Nov 21 2018

[n: n in [1..1600] | Intseq(n)[#Intseq(n)] eq 6]; // Vincenzo Librandi, Nov 24 2018

seq(seq(6*10^n+k, k=0..10^n-1),n=0..3); # Robert Israel, May 08 2017

Select[Range[1000], IntegerDigits[#][[1]] == 6 &] (* T. D. Noe, Oct 02 2012 *)

isok(n) = digits(n)[1] == 6; \\ Michel Marcus, May 08 2017

def A217399(n): return n+(53*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (53*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A217397 Numbers starting with 4.

Original entry on oeis.org

4, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/36 and 2/9, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000867, A011534.
Subsequences include: A045710, A077329, A077680, A106414, A106424.
Cf. A131835, A217394, A217395, A217398, A217399, A217400, A217401, A217402.

Select[Range[1000], IntegerDigits[#][[1]] == 4 &] (* T. D. Noe, Oct 02 2012 *)

def A217397(n): return n+(35*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (35*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A217400 Numbers starting with 7.

Original entry on oeis.org

7, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/63 and 5/36, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000870, A011537.
Subsequences include: A045713, A077332, A077683, A106417, A106427.
Cf. A131835, A217394, A217395, A217397, A217398, A217399, A217401, A217402.

Select[Range[1000], IntegerDigits[#][[1]] == 7 &] (* T. D. Noe, Oct 02 2012 *)

def A217400(n): return n+(62*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (62*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A217401 Numbers starting with 8.

Original entry on oeis.org

8, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/72 and 10/81, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000873, A011538.
Subsequences include: A045714, A077333, A106418, A106428.
Cf. A131835, A217394, A217395, A217397, A217398, A217399, A217400, A217402.

Select[Range[1000], IntegerDigits[#][[1]] == 8 &] (* T. D. Noe, Oct 02 2012 *)

def A217401(n): return n+(71*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (71*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A217402 Numbers starting with 9.

Original entry on oeis.org

9, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/81 and 1/9, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A000981, A011539.
Subsequences include: A077685, A045715, A106419, A106429, A077334.
Cf. A131835, A217394, A217395, A217397, A217398, A217399, A217400, A217401.

Select[Range[1000], IntegerDigits[#][[1]] == 9 &] (* T. D. Noe, Oct 02 2012 *)

def A217402(n): return n+(80*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

a(n) = n + (80*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

A217395 Numbers starting with 3.

Original entry on oeis.org

3, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/27 and 5/18, respectively. - Amiram Eldar, Feb 27 2021

Jeremy Gardiner, Table of n, a(n) for n = 1..1111
Index entries for 10-automatic sequences.

Cf. A006092, A011533.
Subsequences include: A045709, A077328, A077679, A106413, A106423.
Cf. A131835, A217394, A217397, A217398, A217399, A217400, A217401, A217402.

Select[Range[1000], IntegerDigits[#][[1]] == 3 &] (* T. D. Noe, Oct 02 2012 *)

a(n) = n + 26*10^logint(9*n,10)\9; \\ Kevin Ryde, Mar 30 2021

def agen():
  yield 3
  digits, adder = 1, 30
  while True:
    for i in range(10**digits): yield adder + i
    digits, adder = digits+1, adder*10
g = agen()
print([next(g) for i in range(54)]) # Michael S. Branicky, Mar 30 2021

def A217395(n): return n+26*10**(len(str(9*n-8))-1)//9 # Chai Wah Wu, Sep 11 2024

a(n) = n + (26*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 16 2023

A385700 Numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 4.

Original entry on oeis.org

0, 4, 8, 21, 23, 25, 27, 29, 40, 42, 44, 46, 48, 61, 63, 65, 67, 69, 80, 82, 84, 86, 88, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 255, 257, 259, 261, 263, 265, 267, 269

Offset: 1

Stefano Spezia, Jul 07 2025

			263 is a term since 632 = 158*4 is divisible by 4.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Similar sequences for k=1..9: A001477, A273892, A008585, this sequence, A217398, A385701, A385702, A385703, A008591.

Select[Range[0,270],Divisible[FromDigits[RotateLeft[IntegerDigits[#]]],4] &]

isok(k) = if (k==0, return(1)); my(d=digits(k), v = vector(#d-1, i, d[i+1])); v = concat(v, d[1]); fromdigits(v) % 4 == 0; \\ Michel Marcus, Jul 08 2025

def ok(n): return int((s:=str(n))[1:]+s[0])%4 == 0
print([k for k in range(270) if ok(k)]) # Michael S. Branicky, Jul 08 2025

A385701 Numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 6.

Original entry on oeis.org

0, 6, 21, 24, 27, 42, 45, 48, 60, 63, 66, 69, 81, 84, 87, 201, 204, 207, 210, 213, 216, 219, 222, 225, 228, 231, 234, 237, 240, 243, 246, 249, 252, 255, 258, 261, 264, 267, 270, 273, 276, 279, 282, 285, 288, 291, 294, 297, 402, 405, 408, 411, 414, 417, 420, 423, 426, 429

Offset: 1

Stefano Spezia, Jul 07 2025

			426 is a term since 264 = 44*6 is divisible by 6.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Similar sequences for k=1..9: A001477, A273892, A008585, A385700, A217398, this sequence, A385702, A385703, A008591.

Select[Range[0,430],Divisible[FromDigits[RotateLeft[IntegerDigits[#]]],6] &]

isok(k) = if (k==0, return(1)); my(d=digits(k), v = vector(#d-1, i, d[i+1])); v = concat(v, d[1]); fromdigits(v) % 6 == 0; \\ Michel Marcus, Jul 08 2025

def ok(n): return int((s:=str(n))[1:]+s[0])%6 == 0
print([k for k in range(430) if ok(k)]) # Michael S. Branicky, Jul 08 2025

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A131835 Numbers starting with 1.

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A217394 Numbers starting with 2.

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A217399 Numbers starting with 6.

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A217397 Numbers starting with 4.

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A217400 Numbers starting with 7.

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A217401 Numbers starting with 8.

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A217402 Numbers starting with 9.

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