A054054 -id:A054054 - cp's OEIS frontend

A115353 The mode of the digits of n (using smallest mode if multimodal).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 0, 1, 2, 3, 4, 5, 6, 6, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 7, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 0, 0, 0

Offset: 0

Rick L. Shepherd, Jan 21 2006

a(101)=1 and A054054(101)=0, but all previous terms are equivalent.

			a(12)=1 because 1, 2, the digits of 12, each occur the same number of times and 1 is the smaller of the two modes.
a(101)=1 because 1 is the unique mode of 1, 0, 1 (occurring twice while 0 appears only once).

Bence Bernáth, Table of n, a(n) for n = 0..10000

Cf. A054054 (Smallest digit of n).

function nth_term=A115353(n)
     nth_term=mode((num2str(n)-'0'));
end
sequence = arrayfun(@A115353, linspace(0,105,106))
% Bence Bernáth, Jan 06 2023

a[n_] := Min[Commonest[IntegerDigits[n]]]; Array[a,105,0] (* Stefano Spezia, Jan 08 2023 *)

from statistics import mode
def a(n): return int(mode(sorted(str(n))))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jan 08 2023

A284063 Numbers whose smallest decimal digit is 2.

Original entry on oeis.org

2, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92, 222, 223, 224, 225, 226, 227, 228, 229, 232, 233, 234, 235, 236, 237, 238, 239, 242, 243, 244, 245, 246, 247, 248, 249, 252, 253, 254, 255, 256, 257, 258, 259, 262, 263, 264, 265, 266, 267, 268

Offset: 1

Jaroslav Krizek, Mar 19 2017

Numbers n such that A054054(n) = 2.

Prime terms are in A106102.

Cf. Sequences of numbers whose smallest decimal digit is k (for k = 0..9): A011540 (k = 0), A284062 (k = 1), this sequence (k = 2), A284064 (k = 3), A284065 (k = 4), A284066 (k = 5), A284067 (k = 6), A284068 (k = 7), A284069 (k = 8), A002283 (k = 9).

[n: n in [1..100000] | Minimum(Setseq(Set(Sort(&cat[Intseq(n)])))) eq 2]

Select[Range[300], Min[IntegerDigits[#]] == 2 &] (* Alonso del Arte, Mar 19 2017 *)

isok(n) = vecmin(digits(n)) == 2; \\ Michel Marcus, Mar 25 2017

def ok(n): return '2' == min(str(n))
print([m for m in range(269) if ok(m)]) # Michael S. Branicky, Feb 22 2021

A284065 Numbers whose smallest decimal digit is 4.

Original entry on oeis.org

4, 44, 45, 46, 47, 48, 49, 54, 64, 74, 84, 94, 444, 445, 446, 447, 448, 449, 454, 455, 456, 457, 458, 459, 464, 465, 466, 467, 468, 469, 474, 475, 476, 477, 478, 479, 484, 485, 486, 487, 488, 489, 494, 495, 496, 497, 498, 499, 544, 545, 546, 547, 548, 549, 554

Offset: 1

Jaroslav Krizek, Mar 19 2017

Numbers n such that A054054(n) = 4.

Prime terms are in A106104.

Cf. Sequences of numbers whose smallest decimal digit is k (for k = 0..9): A011540 (k = 0), A284062 (k = 1), A284063 (k = 2), A284064 (k = 3), this sequence (k = 4), A284066 (k = 5), A284067 (k = 6), A284068 (k = 7), A284069 (k = 8), A002283 (k = 9).

[n: n in [1..100000] | Minimum(Setseq(Set(Sort(&cat[Intseq(n)])))) eq 4]

With[{k = 4}, Select[Range@ 554, And[Total@ Take[#, k] == 0, #[[k + 1]] > 0] &@ RotateRight@ DigitCount@ # &]] (* Michael De Vlieger, Mar 20 2017 *)
(* or *)
Select[Range[1000], Min[IntegerDigits[#]] == 4 &] (* Giovanni Resta, Mar 22 2017 *)

isok(n) = vecmin(digits(n)) == 4; \\ Michel Marcus, Mar 25 2017

A284066 Numbers whose smallest decimal digit is 5.

Original entry on oeis.org

5, 55, 56, 57, 58, 59, 65, 75, 85, 95, 555, 556, 557, 558, 559, 565, 566, 567, 568, 569, 575, 576, 577, 578, 579, 585, 586, 587, 588, 589, 595, 596, 597, 598, 599, 655, 656, 657, 658, 659, 665, 675, 685, 695, 755, 756, 757, 758, 759, 765, 775, 785, 795, 855

Offset: 1

Jaroslav Krizek, Mar 23 2017

Numbers n such that A054054(n) = 5.

Prime terms are in A106105.

Cf. Sequences of numbers whose smallest decimal digit is k (for k = 0..9): A011540 (k = 0), A284062 (k = 1), A284063 (k = 2), A284064 (k = 3), A284065 (k = 4), this sequence (k = 5), A284067 (k = 6), A284068 (k = 7), A284069 (k = 8), A002283 (k = 9).

[n: n in [1..100000] | Minimum(Setseq(Set(Sort(&cat[Intseq(n)])))) eq 5]

Select[Range[1000], Min[IntegerDigits[#]] == 5 &] (* Giovanni Resta, Mar 23 2017 *)

isok(n) = vecmin(digits(n)) == 5; \\ Michel Marcus, Mar 25 2017

A284067 Numbers whose smallest decimal digit is 6.

Original entry on oeis.org

6, 66, 67, 68, 69, 76, 86, 96, 666, 667, 668, 669, 676, 677, 678, 679, 686, 687, 688, 689, 696, 697, 698, 699, 766, 767, 768, 769, 776, 786, 796, 866, 867, 868, 869, 876, 886, 896, 966, 967, 968, 969, 976, 986, 996, 6666, 6667, 6668, 6669, 6676, 6677, 6678

Offset: 1

Jaroslav Krizek, Mar 23 2017

Numbers n such that A054054(n) = 6.

Prime terms are in A106106.

Cf. Sequences of numbers whose smallest decimal digit is k (for k = 0..9): A011540 (k = 0), A284062 (k = 1), A284063 (k = 2), A284064 (k = 3), A284065 (k = 4), A284066 (k = 5), this sequence (k = 6), A284068 (k = 7), A284069 (k = 8), A002283 (k = 9).

[n: n in [1..100000] | Minimum(Setseq(Set(Sort(&cat[Intseq(n)])))) eq 6]

Select[Range[1000], Min[IntegerDigits[#]] == 6 &] (* Giovanni Resta, Mar 23 2017 *)

isok(n) = vecmin(digits(n)) == 6; \\ Michel Marcus, Mar 25 2017

A284068 Numbers whose smallest decimal digit is 7.

Original entry on oeis.org

7, 77, 78, 79, 87, 97, 777, 778, 779, 787, 788, 789, 797, 798, 799, 877, 878, 879, 887, 897, 977, 978, 979, 987, 997, 7777, 7778, 7779, 7787, 7788, 7789, 7797, 7798, 7799, 7877, 7878, 7879, 7887, 7888, 7889, 7897, 7898, 7899, 7977, 7978, 7979, 7987, 7988, 7989

Offset: 1

Jaroslav Krizek, Mar 23 2017

Numbers n such that A054054(n) = 7.

Prime terms are in A106107.

Cf. Sequences of numbers whose smallest decimal digit is k (for k = 0..9): A011540 (k = 0), A284062 (k = 1), A284063 (k = 2), A284064 (k = 3), A284065 (k = 4), A284066 (k = 5), A284067 (k = 6), this sequence (k = 7), A284069 (k = 8), A002283 (k = 9).

[n: n in [1..100000] | Minimum(Setseq(Set(Sort(&cat[Intseq(n)])))) eq 7]

Select[Range[8000], Min[IntegerDigits[#]] == 7 &] (* Giovanni Resta, Mar 23 2017 *)

isok(n) = vecmin(digits(n)) == 7; \\ Michel Marcus, Mar 25 2017

A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 1, 11, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 0, 2, 1, 2, 2, 22, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 0, 3, 1, 3, 2, 3, 3, 33, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 0, 4, 1, 4, 2, 4, 3, 4, 4, 44, 4, 5, 4, 6, 4

Offset: 0

Reinhard Zumkeller, Sep 14 2015

Length of row n = A262190(n);
T(n,0) = A054054(n);
T(n,A262190(n)-1) = A047813(n).

			.     n |  T(n,*)           n |  T(n,*)              n |  T(n,*)
.  -----+-----------    ------+-------------    -------+--------------
.   100 |  0,1           1000 |  0,1             10000 |  0,1
.   101 |  0,1,101       1001 |  0,1,1001        10001 |  0,1,10001
.   102 |  0,1,2         1002 |  0,1,2           10002 |  0,1,2
.   103 |  0,1,3         1003 |  0,1,3           10003 |  0,1,3
.   104 |  0,1,4         1004 |  0,1,4           10004 |  0,1,4
.   105 |  0,1,5         1005 |  0,1,5           10005 |  0,1,5
.   106 |  0,1,6         1006 |  0,1,6           10006 |  0,1,6
.   107 |  0,1,7         1007 |  0,1,7           10007 |  0,1,7
.   108 |  0,1,8         1008 |  0,1,8           10008 |  0,1,8
.   109 |  0,1,9         1009 |  0,1,9           10009 |  0,1,9
.   110 |  0,1,11        1010 |  0,1,101         10010 |  0,1,1001
.   111 |  1,11,111      1011 |  0,1,11,101      10011 |  0,1,11,1001
.   112 |  1,2,11        1012 |  0,1,2,101       10012 |  0,1,2,1001
.   113 |  1,3,11        1013 |  0,1,3,101       10013 |  0,1,3,1001
.   114 |  1,4,11        1014 |  0,1,4,101       10014 |  0,1,4,1001
.   115 |  1,5,11        1015 |  0,1,5,101       10015 |  0,1,5,1001
.   116 |  1,6,11        1016 |  0,1,6,101       10016 |  0,1,6,1001
.   117 |  1,7,11        1017 |  0,1,7,101       10017 |  0,1,7,1001
.   118 |  1,8,11        1018 |  0,1,8,101       10018 |  0,1,8,1001
.   119 |  1,9,11        1019 |  0,1,9,101       10019 |  0,1,9,1001
.   120 |  0,1,2         1020 |  0,1,2           10020 |  0,1,2
.   121 |  1,2,121       1021 |  0,1,2           10021 |  0,1,2
.   122 |  1,2,22        1022 |  0,1,2,22        10022 |  0,1,2,22
.   123 |  1,2,3         1023 |  0,1,2,3         10023 |  0,1,2,3
.   124 |  1,2,4         1024 |  0,1,2,4         10024 |  0,1,2,4
.   125 |  1,2,5         1025 |  0,1,2,5         10025 |  0,1,2,5  .

Reinhard Zumkeller, Rows n = 0..10000 of triangle, flattened
Index entries for sequences related to palindromes

Cf. A262190 (row lengths), A054054 (left edge), A047813 (right edge), A136522, A002113.

Cf. A031298, A218978.

import Data.List (inits, tails, nub, sort)
a262188 n k = a262188_tabf !! n !! k
a262188_row n = a262188_tabf !! n
a262188_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
   filter (\xs -> length xs == 1 || last xs > 0 && reverse xs == xs) .
          concatMap (tail . inits) . tails) a031298_tabf

A262188_row(n,b=10)=Set(concat(vector(logint(n+!n,b)+1,m,m=n\=b^(m>1);select(is_A002113,vector(logint(m+!m,b)+1,k,m%b^k))))) \\ M. F. Hasler, Jun 19 2018

A115300 Greatest digit of n * least digit of n.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8

Offset: 1

Rick L. Shepherd, Jan 20 2006

a(101) = 0 and A111707(101) = 1, but all previous terms match.

a(n) = A169669(n) for n <= 100.

			a(3) = 3 * 3 = 9, a(232) = 3 * 2 = 6, a(1889009898) = 9 * 0 = 0.

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

Cf. A037904 (greatest-least), A115299 (greatest+least), A111707.

Cf. A054054, A054055, A169669.

a115300 n = a054054 n * a054055 n  -- Reinhard Zumkeller, Apr 29 2015

Array[Max[#] * Min[#] &@ IntegerDigits[#] &, 81] (* James C. McMahon, Aug 18 2024 *)

a(n) = my(d=digits(n)); vecmin(d)*vecmax(d); \\ Michel Marcus, Aug 18 2024

def a(n): d = list(map(int, str(n))); return max(d) * min(d)
print([a(n) for n in range(1, 82)]) # Michael S. Branicky, Dec 12 2023

a(n) = A054054(n)*A054055(n). - Reinhard Zumkeller, Apr 29 2015

A072544 Numbers whose smallest decimal digit is also the initial digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124

Offset: 1

Reinhard Zumkeller, Aug 04 2002

A054054(a(n)) = A000030(a(n));

the sequence differs from A009994, A032857 and A032898: a(65)=121 is not in A009994, a(58)=113 is not in A032857 and a(56)=111 is not in A032898.

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

Cf. A072543, A009994.

a072544 n = a072544_list !! (n-1)
a072544_list = [x | x <- [0..], a054054 x == a000030 x]
-- Reinhard Zumkeller, Apr 25 2012

sddiQ[n_]:=Module[{idn=IntegerDigits[n]},First[idn]==Min[idn]]; Select[ Range[ 0,130],sddiQ] (* Harvey P. Dale, Oct 30 2011 *)

A097385 a(n) = (largest digit of n)^(smallest digit of n) + n.

Original entry on oeis.org

1, 2, 6, 30, 260, 3130, 46662, 823550, 16777224, 387420498, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 21, 23, 26, 32, 40, 50, 62, 76, 92, 110, 31, 34, 41, 60, 98, 160, 252, 380, 550, 768, 41, 45, 58, 107, 300, 670, 1342, 2448, 4144, 6610, 51, 56, 77, 178, 679, 3180

Offset: 0

Jason Earls, Aug 18 2004

			a(2345) = 2370 because 5^2 + 2345 = 2370.

Mia Boudreau, Table of n, a(n) for n = 0..10000

Cf. A054054, A054055, A097386, A097387.

def a(n): return int(max(s:=str(n)))**int(min(s)) + n
print([a(n) for n in range(56)]) # Michael S. Branicky, Jul 21 2025

a(n) = A054055(n)^A054054(n) + n. - Mia Boudreau, Jul 17 2025

a(0) corrected and 2 terms merged by Mia Boudreau, Jul 16 2025

cp's OEIS Frontend

A115353 The mode of the digits of n (using smallest mode if multimodal).

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A284063 Numbers whose smallest decimal digit is 2.

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A284065 Numbers whose smallest decimal digit is 4.

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A284066 Numbers whose smallest decimal digit is 5.

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A284067 Numbers whose smallest decimal digit is 6.

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A284068 Numbers whose smallest decimal digit is 7.

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A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

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A115300 Greatest digit of n * least digit of n.

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