A115353 -id:A115353 - cp's OEIS frontend

A115516 The mode of the bits of n (using 0 if bimodal).

0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0

Offset: 0

Rick L. Shepherd, Jan 23 2006

If n is a term of A044951, A115517(n) = a(n); otherwise, A115517(n) = 1 and a(n) = 0 (and n is a term of A031443).

			a(5)=1 because 5 = 101 (binary) and 0 occurs once, but 1 occurs twice, so 1 is the mode. 5 is a member of A044951 (Numbers with no two equally numerous base 2 digits).
a(10)=0 because 10 = 1010 (binary), where 0 and 1 each occur twice. As these bits are bimodal, 0 is chosen. 10 is a member of A031443 (Digitally balanced numbers: numbers which in base 2 have the same number of 0's as 1's.).

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A115517 (same but use 1 if bimodal), A031443 (n's bits bimodal), A044951 (n's bits unimodal), A115353 (mode of n's decimal digits).

Array[Min[Commonest[IntegerDigits[#, 2]]] &, 100, 0] (* Paolo Xausa, May 21 2024 *)

{for(n=0,104, b=binary(n); l=length(b); s=sum(m=1,l,b[m]); if(s>l-s, a=1, a=0); print1(a,","))}

a(A031443(k))=0 for k>=1.

A115517 The mode of the bits of n (using 1 if bimodal).

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0

Offset: 0

Rick L. Shepherd, Jan 23 2006

If n is a term of A044951, A115516(n) = a(n); otherwise, A115516(n) = 0 and a(n) = 1 (and n is a term of A031443).

			a(5)=1 because 5 = 101 (binary) and 0 occurs once, but 1 occurs twice, so 1 is the mode. 5 is a member of A044951 (Numbers with no two equally numerous base 2 digits).
a(10)=1 because 10 = 1010 (binary), where 0 and 1 each occur twice. As these bits are bimodal, 1 is chosen. 10 is a member of A031443 (Digitally balanced numbers: numbers which in base 2 have the same number of 0's as 1's.).

Paolo Xausa, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Plot a(n) at (x,y) = (n mod 1024, floor(n/1024)), n = 1..1048576, with black indicating 0 and white 1.

Cf. A115516 (same but use 0 if bimodal), A031443 (n's bits bimodal), A044951 (n's bits unimodal), A115353 (mode of n's decimal digits).

{0}~Join~Table[If[DigitCount[n, 2, 0] > DigitCount[n, 2, 1], 0, 1], {n, 120}] (* Harvey P. Dale, Jul 29 2019 *)
Array[Max[Commonest[IntegerDigits[#, 2]]] &, 100, 0] (* Paolo Xausa, May 21 2024 *)

{for(n=0,104, b=binary(n); l=length(b); s=sum(m=1,l,b[m]); if(s>=l-s, a=1, a=0); print1(a,","))}

def a(n): return int(n.bit_count() >= ((n.bit_length()+1)>>1)) if n else 0
print([a(n) for n in range(105)]) # Michael S. Branicky, May 21 2024

a(A031443(k))=1 for k>=1.

A373064 The mode of the digits of n (using largest mode if multimodal).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, May 21 2024

First differs from A054055 at n = 100.

			a(100) = 0 because 0 is the digit occurring with the highest frequency in 100.
a(123300) = 3 because both 0 and 3 occur with the (same) highest frequency in 123300, and 3 is the largest digit.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A054055, A115353.

Array[Max[Commonest[IntegerDigits[#]]] &, 120, 0]

from statistics import multimode
def a(n): return int(max(multimode(str(n))))
print([a(n) for n in range(105)]) # Michael S. Branicky, May 21 2024

A379180 Nonnegative integers with mode and mean of the digits equal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1012, 1021, 1102, 1111, 1120, 1201, 1210, 1223, 1232, 1322, 1335, 1353, 1447, 1474, 1533, 1559, 1595, 1744, 1955, 2011, 2024, 2042, 2101, 2110, 2123, 2132

Offset: 1

Stefano Spezia, Dec 17 2024

			1021 is a term since the mode and the mean of the digits are equal to 1.

Cf. A115353, A378560, A378835, A379181.

Select[Range[0,2200], Commonest[IntegerDigits[#]] == {Mean[IntegerDigits[#]]} &]

A383306 Nonnegative integers whose difference between the largest and smallest digits is equal to the mode of its digits.

Original entry on oeis.org

0, 101, 110, 112, 121, 202, 211, 220, 224, 242, 303, 330, 336, 363, 404, 422, 440, 448, 484, 505, 550, 606, 633, 660, 707, 770, 808, 844, 880, 909, 990, 1011, 1022, 1033, 1044, 1055, 1066, 1077, 1088, 1099, 1101, 1110, 1112, 1121, 1202, 1211, 1220, 1223, 1232

Offset: 1

Stefano Spezia, Apr 22 2025

It includes only terms with unimodal digits.

			363 is a term since 6 - 3 = 3 is equal to the mode of {3, 3, 6}.

Cf. A037904, A054054, A054055, A115353, A383307.

Select[Range[0,1400], {Max[d=IntegerDigits[#]]-Min[d]}==Commonest[d] &]

A383307 a(n) is number of n-digit nonnegative integers whose difference between the largest and smallest digits is equal to the mode of its digits.

Original entry on oeis.org

1, 0, 30, 631, 8318, 84939, 762621, 6836799, 66714966, 698183347, 7345264685, 74862560359, 738289921745, 7152117119827, 69258386123495, 678852874461343, 6757612542040310, 67956663939884115, 684414144298352061, 6858156111567293583, 68247431544857431593, 675967074881581484903

Offset: 1

Stefano Spezia, Apr 22 2025

Wikipedia, Mode (statistics)

Cf. A037904, A054054, A054055, A115353, A383306.

a[n_]:=Module[{c=KroneckerDelta[n,1]}, For[k=10^(n-1), k<=10^n, k++, If[{Max[d=IntegerDigits[k]]-Min[d]}==Commonest[d], c++]]; c]; Array[a,6]

Conjecture: lim_{n->oo} a(n+1)/a(n) = 10. - Stefano Spezia, Apr 26 2025

More terms from Bert Dobbelaere, Apr 25 2025

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A115516 The mode of the bits of n (using 0 if bimodal).

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A115517 The mode of the bits of n (using 1 if bimodal).

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A373064 The mode of the digits of n (using largest mode if multimodal).

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A379180 Nonnegative integers with mode and mean of the digits equal.

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A383306 Nonnegative integers whose difference between the largest and smallest digits is equal to the mode of its digits.

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A383307 a(n) is number of n-digit nonnegative integers whose difference between the largest and smallest digits is equal to the mode of its digits.

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