A217399 -id:A217399 - 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

A217398 Numbers starting with 5.

Original entry on oeis.org

5, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542

Offset: 1

Jeremy Gardiner, Oct 02 2012

The lower and upper asymptotic densities of this sequence are 1/45 and 5/27, respectively. - Amiram Eldar, Feb 27 2021

Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 5. - Stefano Spezia, Jul 08 2025

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

Cf. A000867, A011535.
Subsequences include: A045711, A077330, A077681, A106415, A106425.
Cf. A131835, A217394, A217395, A217397, A217399, A217400, A217401, A217402.

a217398 n = a217398_list !! (n-1)
a217398_list = filter ((== 5) . a000030) [1..]
-- Reinhard Zumkeller, Mar 13 2014

Select[Range[1000], IntegerDigits[#][[1]] == 5 &] (* T. D. Noe, Oct 02 2012 *)

def A217398(n): return n+(44*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

A000030(a(n)) = 5; A143473(a(n)) = a(n). - Reinhard Zumkeller, Mar 13 2014

a(n) = n + (44*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

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

A077682 Squarefree numbers beginning with 6.

Original entry on oeis.org

6, 61, 62, 65, 66, 67, 69, 601, 602, 606, 607, 609, 610, 611, 613, 614, 615, 617, 618, 619, 622, 623, 626, 627, 629, 631, 633, 634, 635, 638, 641, 642, 643, 645, 646, 647, 649, 651, 653, 654, 655, 658, 659, 661, 662, 663, 665, 667, 669, 670, 671, 673, 674

Offset: 1

Amarnath Murthy, Nov 16 2002

Intersection of A005117 and A217399. - Michel Marcus, Sep 14 2013

Lower density is 1/(10*Pi^2), upper density is 20/(21*Pi^2). - Charles R Greathouse IV, Nov 05 2017

Muniru A Asiru, Table of n, a(n) for n = 1..6761

Cf. A077677, A077678, A077679, A077680, A077681, A077683, A077684, A077685.

Cf. A005117, A217399.

with(numtheory): select(issqrfree,[seq(seq(6*10^n+k,k=0..10^n-1),n=0..3)]); # Muniru A Asiru, Nov 22 2018

Select[Range[1000],SquareFreeQ[#]&&First[IntegerDigits[#]]==6&] (* Harvey P. Dale, May 19 2012 *)

is(n)=n>5 && digits(n)[1]==6 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017

More terms from Sascha Kurz, Jan 28 2003

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

A043041 Base-10 palindromes that start with 6.

Original entry on oeis.org

6, 66, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 6006, 6116, 6226, 6336, 6446, 6556, 6666, 6776, 6886, 6996, 60006, 60106, 60206, 60306, 60406, 60506, 60606, 60706, 60806, 60906, 61016, 61116, 61216, 61316, 61416, 61516, 61616, 61716, 61816, 61916, 62026

Offset: 1

Clark Kimberling

Jaroslav Krizek and David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2222 terms from Jaroslav Krizek)

Intersection of A002113 and A217399.

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]]; Select[Range[0, 100000], IntegerDigits[#][[1]] == 6 && palQ[#, 10] &] (* T. D. Noe, Mar 12 2013 *)
Select[Flatten[Table[Range[6*10^n,7*10^n-1],{n,0,4}]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 25 2017 *)

nxt2113(n)={my(d=digits(n)); i=(#d+1)\2; while(i&&d[i]==9, d[i]=0; d[#d+1-i]=0; i--); if(i, d[i]++; d[#d+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1); d}
nxt(n) = my(d = nxt2113(n)); if(d[1]==6, fromdigits(d), 6*(10^#d+1))
print1(a = 6, ", "); for (n=1, 30, print1(a = nxt(a), ", ")) \\ David A. Corneth, Jun 25 2017

a(n) = A002113(k) iff k starts with 7 or 16. - David A. Corneth, Jun 25 2017

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A131835 Numbers starting with 1.

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A217398 Numbers starting with 5.

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A217394 Numbers starting with 2.

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A077682 Squarefree numbers beginning 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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