A065881 -id:A065881 - cp's OEIS frontend

A277593 Numbers k such that k/10^m == 6 mod 10, where 10^m is the greatest power of 10 that divides n.

6, 16, 26, 36, 46, 56, 60, 66, 76, 86, 96, 106, 116, 126, 136, 146, 156, 160, 166, 176, 186, 196, 206, 216, 226, 236, 246, 256, 260, 266, 276, 286, 296, 306, 316, 326, 336, 346, 356, 360, 366, 376, 386, 396, 406, 416, 426, 436, 446, 456, 460, 466, 476, 486

Offset: 1

Clark Kimberling, Nov 07 2016

Positions of 6 in A065881.

Numbers having 6 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277588-A277592, A277594-A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)

is(n)=n && n/10^valuation(n,10)%10==6 \\ Charles R Greathouse IV, Jan 31 2017

A277589 Numbers k such that k/10^m == 2 mod 10, where 10^m is the greatest power of 10 that divides n.

Original entry on oeis.org

2, 12, 20, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 120, 122, 132, 142, 152, 162, 172, 182, 192, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 302, 312, 320, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 420, 422, 432, 442, 452, 462, 472

Offset: 1

Clark Kimberling, Nov 05 2016

Positions of 2 in A065881.

Numbers having 2 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277588, A277590-A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)

is(n)=n && n/10^valuation(n,10)%10==2 \\ Charles R Greathouse IV, Jan 31 2017

A277591 Numbers k such that k/10^m == 4 mod 10, where 10^m is the greatest power of 10 that divides n.

Original entry on oeis.org

4, 14, 24, 34, 40, 44, 54, 64, 74, 84, 94, 104, 114, 124, 134, 140, 144, 154, 164, 174, 184, 194, 204, 214, 224, 234, 240, 244, 254, 264, 274, 284, 294, 304, 314, 324, 334, 340, 344, 354, 364, 374, 384, 394, 400, 404, 414, 424, 434, 440, 444, 454, 464, 474

Offset: 1

Clark Kimberling, Nov 05 2016

Positions of 4 in A065881.

Numbers having 4 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277588-A277590, A277592-A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)

A277592 Numbers k such that k/10^m == 5 mod 10, where 10^m is the greatest power of 10 that divides n.

Original entry on oeis.org

5, 15, 25, 35, 45, 50, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 150, 155, 165, 175, 185, 195, 205, 215, 225, 235, 245, 250, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 350, 355, 365, 375, 385, 395, 405, 415, 425, 435, 445, 450, 455, 465, 475, 485

Offset: 1

Clark Kimberling, Nov 05 2016

Positions of 5 in A065881.
Numbers having 5 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.
This sequence differs from A121025, which includes 540.

Cf. A277588-A277591, A277593-A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)

A277595 Numbers k such that k/10^m == 8 mod 10, where 10^m is the greatest power of 10 that divides k.

Original entry on oeis.org

8, 18, 28, 38, 48, 58, 68, 78, 80, 88, 98, 108, 118, 128, 138, 148, 158, 168, 178, 180, 188, 198, 208, 218, 228, 238, 248, 258, 268, 278, 280, 288, 298, 308, 318, 328, 338, 348, 358, 368, 378, 380, 388, 398, 408, 418, 428, 438, 448, 458, 468, 478, 480, 488

Offset: 1

Clark Kimberling, Nov 07 2016

Positions of 8 in A065881.

Numbers having 8 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277588-A277594, A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)
fQ[n_]:=Module[{sp=Split[IntegerDigits[n]]},If[MemberQ[sp[[-1]],0],sp = Drop[ sp, -1]];MemberQ[sp[[-1]],8]]; Select[Range[500],fQ] (* Harvey P. Dale, Sep 14 2018 *)

is(n)=n && n/10^valuation(n,10)%10==6 \\ Charles R Greathouse IV, Jan 31 2017

A065882 Ultimate modulo 4: right-hand nonzero digit of n when written in base 4.

Original entry on oeis.org

1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, 2, 1, 2, 3, 3, 1, 2, 3, 2, 1, 2, 3, 1, 1, 2, 3, 2, 1

Offset: 1

Henry Bottomley, Nov 26 2001

From Bradley Klee, Sep 12 2015: (Start)
In some guise, this sequence is a linear encoding of the three fixed-point half-hex tilings (cf. Baake & Grimm, Frettlöh). Applying a permutation, morphism x -> 123x becomes x -> x123, which has three fixed points. Applying a partition, morphism x -> x123 becomes x ->{{3,2},{x,1}} or
            3 2 3 2
            3 1 2 1
     3 2    3 2 3 2
x -> x 1 -> x 1 1 1 -> etc.,
which is the substitution rule for the half-hex tiling when the numbers 1,2,3 determine the direction of a dissecting diameter inscribed on each hexagon.
(End)

			a(7)=3 and a(112)=3, since 7 is written in base 4 as 13 and 112 as 1300.

M. Baake and U. Grimm, Aperiodic Order Vol. 1, Cambridge University Press, 2013, page 205.

Harry J. Smith, Table of n, a(n) for n = 1..1000
D. Frettlöh, Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor, Dissertation, Universität Dortmund, 2002.
Index entries for sequences related to final digits of numbers
Index entries for sequences that are fixed points of mappings

In base 2 this is A000012, base 3 A060236 and base 10 A065881.
Defining relations for g.f. similar to A014577.
Cf. A010873, A037898, A065883, A190593.

f:= proc(n)
local x:=n;
   while x mod 4 = 0 do x:= x/4 od:
   x mod 4;
end proc;
map(f, [$1..100]); # Robert Israel, Jan 05 2016

Nest[ Flatten[ # /. {1 -> {1, 2, 3, 1}, 2 -> {1, 2, 3, 2}, 3 -> {1, 2, 3, 3}}] &, {1}, 4] (* Robert G. Wilson v, May 07 2005 *)
b[n_] := CoefficientList[Series[
    With[{f0 = (x + 2 x^2 + 3 x^3)/(1 - x^4)},
     Nest[ (# /. x -> x^4) + f0 &, f0, Ceiling[Log[4, n/3]]]],
{x, 0, n}], x][[2 ;; -1]]; b[100](* Bradley Klee, Sep 12 2015 *)
Table[Mod[n/4^IntegerExponent[n, 4], 4], {n, 1, 120}] (* Clark Kimberling, Oct 19 2016 *)

a(n) = (n/4^valuation(n,4))%4; \\ Joerg Arndt, Sep 13 2015

def A065882(n): return (n>>((~n & n-1).bit_length()&-2))&3 # Chai Wah Wu, Aug 21 2023

If n mod 4 = 0 then a(n) = a(n/4), otherwise a(n) = n mod 4. a(n) = A065883(n) mod 4.
Fixed point of the morphism: 1 ->1231, 2 ->1232, 3 ->1233, starting from a(1) = 1. Sequence read mod 2 gives A035263. a(n) = A007913(n) mod 4. - Philippe Deléham, Mar 28 2004
G.f. g(x) satisfies g(x) = g(x^4) + (x + 2 x^2 + 3 x^3)/(1 - x^4). - Bradley Klee, Sep 12 2015

A162501 Lexicographically earliest permutation of the natural numbers such that in decimal representation the initial digit for each term is equal to the last nonzero digit of its predecessor; a(1)=1.

Original entry on oeis.org

1, 10, 11, 12, 2, 20, 21, 13, 3, 30, 31, 14, 4, 40, 41, 15, 5, 50, 51, 16, 6, 60, 61, 17, 7, 70, 71, 18, 8, 80, 81, 19, 9, 90, 91, 100, 101, 102, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 92, 200, 201, 103, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 93, 300, 301

Offset: 1

Reinhard Zumkeller, Jul 05 2009

A000030(a(n+1)) = A065881(a(n));
inverse of A162502: a(A162502(n)) = A162502(a(n)) = n;
a(a(n)) = A162503(n).

A076654, A130571.

A277588 Numbers k such that k/10^m == 1 mod 10, where 10^m is the greatest power of 10 that divides n.

Original entry on oeis.org

1, 10, 11, 21, 31, 41, 51, 61, 71, 81, 91, 100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 210, 211, 221, 231, 241, 251, 261, 271, 281, 291, 301, 310, 311, 321, 331, 341, 351, 361, 371, 381, 391, 401, 410, 411, 421, 431, 441, 451, 461, 471

Offset: 1

Clark Kimberling, Nov 05 2016

Positions of 1 in A065881.

Numbers having 1 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277589-A277596.

M:= 4: # to get all terms with <= M digits
A:= sort([seq(seq(10^d*(10*x+1),x=0..10^(M-1-d)-1),d=0..M-2)]); # Robert Israel, Nov 07 2016

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)
f[n_] := Block[{m = n}, While[ Mod[m, 10] == 0, m /= 10]; Mod[m, 10]]; Flatten@ Position[ Array[f, 500], 1] (* Robert G. Wilson v, Nov 06 2016 *)

is(n)=n && n/10^valuation(n,10)%10==1 \\ Charles R Greathouse IV, Jan 31 2017

A277590 Numbers k such that k/10^m == 3 mod 10, where 10^m is the greatest power of 10 that divides n.

Original entry on oeis.org

3, 13, 23, 30, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, 130, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 230, 233, 243, 253, 263, 273, 283, 293, 300, 303, 313, 323, 330, 333, 343, 353, 363, 373, 383, 393, 403, 413, 423, 430, 433, 443, 453, 463, 473

Offset: 1

Clark Kimberling, Nov 05 2016

Positions of 3 in A065881.

Numbers having 3 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277588, A277589, A277591-A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)

is(n)=n && n/10^valuation(n,10)%10==3 \\ Charles R Greathouse IV, Jan 31 2017

A277594 Numbers k such that k/10^m == 7 mod 10, where 10^m is the greatest power of 10 that divides n.

Original entry on oeis.org

7, 17, 27, 37, 47, 57, 67, 70, 77, 87, 97, 107, 117, 127, 137, 147, 157, 167, 170, 177, 187, 197, 207, 217, 227, 237, 247, 257, 267, 270, 277, 287, 297, 307, 317, 327, 337, 347, 357, 367, 370, 377, 387, 397, 407, 417, 427, 437, 447, 457, 467, 470, 477, 487

Offset: 1

Clark Kimberling, Nov 07 2016

Positions of 7 in A065881.

Numbers having 7 as rightmost nonzero digit in base 10. This is one sequence in a 10-way splitting of the positive integers; the other nine are indicated in the Mathematica program.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A277588-A277593, A277595-A277596.

z = 460; a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, z}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[10, 1] (* A277588 *)
p[10, 2] (* A277589 *)
p[10, 3] (* A277590 *)
p[10, 4] (* A277591 *)
p[10, 5] (* A277592 *)
p[10, 6] (* A277593 *)
p[10, 7] (* A277594 *)
p[10, 8] (* A277595 *)
p[10, 9] (* A277596 *)

is(n)=n && n/10^valuation(n,10)%10==7 \\ Charles R Greathouse IV, Jan 31 2017

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A277593 Numbers k such that k/10^m == 6 mod 10, where 10^m is the greatest power of 10 that divides n.

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A277589 Numbers k such that k/10^m == 2 mod 10, where 10^m is the greatest power of 10 that divides n.

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A277591 Numbers k such that k/10^m == 4 mod 10, where 10^m is the greatest power of 10 that divides n.

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A277592 Numbers k such that k/10^m == 5 mod 10, where 10^m is the greatest power of 10 that divides n.

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A277595 Numbers k such that k/10^m == 8 mod 10, where 10^m is the greatest power of 10 that divides k.

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A065882 Ultimate modulo 4: right-hand nonzero digit of n when written in base 4.

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A162501 Lexicographically earliest permutation of the natural numbers such that in decimal representation the initial digit for each term is equal to the last nonzero digit of its predecessor; a(1)=1.

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A277588 Numbers k such that k/10^m == 1 mod 10, where 10^m is the greatest power of 10 that divides n.

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