A044821 -id:A044821 - cp's OEIS frontend

A044824 Positive integers having distinct base-13 run lengths.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 169, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 338, 352, 364, 365, 366, 367, 368, 369, 370, 371

Offset: 1

Clark Kimberling

			338 = 200_13 is in the sequence as it has distinct run lengths of distinct digits (2, 1). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

Select[Range[400],Union[Tally[Length/@Split[IntegerDigits[#,13]]][[All,2]]] == {1}&] (* Harvey P. Dale, Sep 15 2020 *)

A044825 Positive integers having distinct base-14 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 196, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 226, 241, 256, 271, 286, 301, 316, 331, 346, 361, 376, 391, 392, 407, 420, 421, 422, 423

Offset: 1

Clark Kimberling

			346 = 1AA_14 is in the sequence as it has distinct run lengths of distinct digits (1, 2). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

A044826 Positive integers having distinct base-15 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 225, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 257, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 450, 466

Offset: 1

Clark Kimberling

			337 = 177_15 is in the sequence as it has distinct run lengths of distinct digits (1, 2). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

A044827 Positive integers having distinct base-16 run lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 256, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 290, 307, 324, 341, 358, 375, 392, 409, 426, 443, 460, 477

Offset: 1

Clark Kimberling

			324 = 144_16 is in the sequence as it has distinct run lengths of distinct digits (1, 2). - _David A. Corneth_, Jan 04 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A044813, A044814, A044815, A044816, A044817, A044818, A044819, A044820, A044821, A044822, A044823, A044824, A044825, A044826, A044827 (base 2 to base 16).

A297148 Numbers whose base-10 digits d(m), d(m-1),..., d(0) have m=0 or else d(i) = d(i+1) for some i in {0,1,...,m-1}.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 200, 211, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 300

Offset: 1

Clark Kimberling, Jan 15 2018

These numbers comprise the complement of the set of numbers in the union of A297146 and A297147.

Differs from A044821 first for 1001, which is in this sequence but not in A044821. - R. J. Mathar, Jan 17 2018

			Base-10 digits of 65536: 6,5,5,3,6, so that 65536 is in the sequence.

Cf. A297146, A297147.

read("transforms") :
isA297148 := proc(n)
    local dgs,ud;
    dgs := convert(n,base,10) ;
    if nops(dgs) < 2 then
        return true;
    end if;
    if 0 in DIFF(dgs) then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 300 do
    if isA297148(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 18 2018

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 10; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297146 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297147 *)
Complement[Range[z], Union[u, v]]  (* A297148 *)

A048321 Reading a(n) expansion from left to right, run lengths strictly decrease.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 440, 441, 442, 443, 444, 445, 446, 447, 448

Offset: 1

Patrick De Geest, Feb 15 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A037013, A043636, A043713, A044821, A044959.

import Data.List (group)
a048321 n = a048321_list !! (n-1)
a048321_list = filter f [0..] where
   f x = all (< 0) $ zipWith (-) (tail zs) zs
         where zs =  map length $ group $ show x
-- Reinhard Zumkeller, May 01 2015

A048307 Numbers whose decimal expansions, read from left to right, have run lengths that strictly increase.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 200, 211, 222, 233, 244, 255, 266, 277, 288, 299, 300, 311, 322, 333, 344, 355, 366, 377, 388, 399, 400, 411, 422, 433, 444, 455, 466, 477, 488

Offset: 0

Patrick De Geest, Feb 15 1999

Cf. A037015, A043636, A043713, A044821, A044959.

A380596 Numbers with embedded palindromes as proper substrings of the term.

Original entry on oeis.org

100, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 200, 211, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 300, 311, 322, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399

Offset: 1

James S. DeArmon, Jan 27 2025

An embedded palindrome is a substring of at least two contiguous digits (since a single digit is trivially a palindrome). E.g., 121 is a palindrome, but has no embedded palindromes; 110 has the embedded palindrome "11".

Alternatively, k contains a proper substring of the form dd or ded, where d and e are single decimal digits (i.e., a length-2 or -3 palindrome). - Michael S. Branicky, Feb 08 2025

			100 is a term, since "00" is a palindrome; 1001 is a term for the same reason.
1020 is a term, since "020" is a palindrome; 10201 is a term for the same reason.

Cf. A002113.

Significant overlap with A044821 for terms below 1000.

from itertools import combinations
def get_all_substrings(string):
    length = len(string) + 1
    return [string[x:y] for x, y in combinations(range(length), r=2)]
def is_palindrome(n):
    return str(n) == str(n)[::-1]
def ok(n):
    subsets = get_all_substrings(str(n))
    subsets = [subset for subset in subsets if is_palindrome(subset) and len(subset)>1 and len(subset)0
print([k for k in range (100,400) if ok(k)])

def ok(n):
    s = str(n)
    return any(p == p[::-1] and len(p) < len(s) for p in (s[i:i+j] for j in (2, 3) for i in range(len(s)-j+1)))
print([k for k in range(400) if ok(k)]) # Michael S. Branicky, Feb 08 2025

cp's OEIS Frontend

A044824 Positive integers having distinct base-13 run lengths.

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Mathematica

A044825 Positive integers having distinct base-14 run lengths.

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A044826 Positive integers having distinct base-15 run lengths.

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A044827 Positive integers having distinct base-16 run lengths.

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A297148 Numbers whose base-10 digits d(m), d(m-1),..., d(0) have m=0 or else d(i) = d(i+1) for some i in {0,1,...,m-1}.

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Maple

Mathematica

A048321 Reading a(n) expansion from left to right, run lengths strictly decrease.

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Haskell

A048307 Numbers whose decimal expansions, read from left to right, have run lengths that strictly increase.

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A380596 Numbers with embedded palindromes as proper substrings of the term.

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Python

Python