A037834 -id:A037834 - cp's OEIS frontend

A297248 Total variation of base-16 digits of n; see Comments.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 5, 4, 3

Offset: 1

Clark Kimberling, Jan 17 2018

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers:

			3^10 in base 16:  14, 6, 10, 9; here, DV = 9 and UV = 4, so that a(2^20) = 13.

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

Cf. A297246, A297247, A297330.

b = 16; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &,      Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, e.g. A037834 *)

A374176 a(n) is the maximum number of consecutive bit changes in the binary representation of n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 1, 3, 2, 1, 2, 1, 0, 1, 1, 2, 1, 3, 4, 2, 2, 1, 1, 3, 2, 1, 2, 1, 0, 1, 1, 2, 1, 2, 3, 1, 1, 3, 3, 5, 4, 2, 2, 2, 2, 1, 1, 2, 1, 3, 4, 2, 2, 1, 1, 3, 2, 1, 2, 1, 0, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 4, 3, 1, 2, 1, 1, 3, 3, 3, 3, 5, 6, 4, 4, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 3, 1, 1, 3, 3

Offset: 1

Hugo Pfoertner, Jul 05 2024

			a(1117) = 3:
  1117_2 = [1 0 0 0 1 0 1 1 1 0 1]
             ^     ^ ^ ^     ^ ^
             1       3        2
            consecutive changes

Hugo Pfoertner, Table of n, a(n) for n = 1..8192

Cf. A005811, A037834, A038554, A038374.

Cf. A000975 (index of first occurrence of n).

b:= n-> `if`(n<2, [0$2], (f-> (t-> [t, max(t, f[2])])(
        `if`(n mod 4 in {0, 3}, 0, f[1]+1)))(b(iquo(n, 2)))):
a:= n-> b(n)[2]:
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 07 2024

a(n) = {my(b=digits(n,2), d=#b, m=0, j=b[1], c=0); for(k=2, d, if(b[k]!=j, c++; m=max(m,c), c=0); j=b[k]); m}

def a(n):
    b, c, m = bin(n)[2:], 0, 0
    for i in range(len(b)-1):
        if b[i] != b[i+1]: c += 1
        else: m = max(m, c); c = 0
    return max(m, c)
print([a(n) for n in range(1, 89)]) # Michael S. Branicky, Jul 06 2024

# using formula
from itertools import groupby
def a(n): return max((len(list(g)) for k, g in groupby(bin(n^(n>>1))[3:]) if k=="1"), default=0)
print([a(n) for n in range(1, 89)]) # Michael S. Branicky, Jul 06 2024

a(n) = A038374(A038554(n)), with A038374(0) = 0. - Michael S. Branicky, Jul 06 2024

A037835 Sum{|d(i)-d(i-1)|: i=0,1,...,m}, where Sum{d(i)*3^i: i=0,1,...,m} is base 3 representation of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 3, 2, 1, 2, 3, 4, 2, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 3, 5, 4, 3, 1, 2, 3, 1, 0, 1, 3, 2, 1, 3, 4, 5, 3, 2, 3, 3, 2, 1, 2, 3, 4, 4, 3, 4, 6, 5, 4, 2, 3, 4, 2, 1, 2, 4, 3, 2, 2, 3, 4, 2, 1, 2, 2, 1, 0, 1, 2, 3, 3, 2, 3, 5, 4, 3, 3

Offset: 1

Clark Kimberling

This is the base-3 total variation sequence; see A297330. - Clark Kimberling

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

Cf. A297330.

A037835 := proc(n)
    local dgs ;
    dgs := convert(n,base,3);
    add( abs(op(i,dgs)-op(i-1,dgs)),i=2..nops(dgs)) ;
end proc: # R. J. Mathar, Oct 16 2015

b = 3; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &,      Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 19 2018

A037836 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-4 total variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

A037836 := proc(n)
    local dgs ;
    dgs := convert(n,base,4);
    add( abs(op(i,dgs)-op(i-1,dgs)),i=2..nops(dgs)) ;
end proc: # R. J. Mathar, Oct 16 2015

b = 4; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 20 2018

A037837 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-5 total variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

A037837 := proc(n)
    local dgs ;
    dgs := convert(n,base,5);
    add( abs(op(i,dgs)-op(i-1,dgs)),i=2..nops(dgs)) ;
end proc: # R. J. Mathar, Oct 16 2015

b = 5; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 20 2018

A037838 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-6 total variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

A037838 := proc(n)
    local dgs ;
    dgs := convert(n,base,6);
    add( abs(op(i,dgs)-op(i-1,dgs)),i=2..nops(dgs)) ;
end proc:
seq(A037838(n),n=1..100) ; # R. J. Mathar, Jul 31 2024

b = 6; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 20 2018

A037839 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-7 total variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

b = 7; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 20 2018

A037840 a(n)=Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*8^i: i=0,1,...,m} is the base 8 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-8 total variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

b = 8; z = 120; t = Table[Total@ Flatten@ Map[Abs@ Differences@ # &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 20 2018

A037841 a(n)=Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*9^i: i=0,1,...,m} is the base 9 representation of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-9 total variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

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

Cf. A297330.

b = 9; z = 120; t = Table[Total@ Flatten@ Map[Abs@ Differences@ # &, Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, A037834 *)

Updated by Clark Kimberling, Jan 20 2018

A297236 Total variation of base-12 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 7, 6, 5

Offset: 1

Clark Kimberling, Jan 17 2018

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See A297330 for a guide to related sequences and partitions of the natural numbers:

			2^20 in base 12:  4, 2, 6, 9, 9, 4; here, DV = 7 and UV = 7, so that a(2^20) = 14.

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

Cf. A297234, A297235, A297330.

b = 12; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &,      Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, e.g. A037834 *)

cp's OEIS Frontend

A297248 Total variation of base-16 digits of n; see Comments.

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A374176 a(n) is the maximum number of consecutive bit changes in the binary representation of n.

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A037835 Sum{|d(i)-d(i-1)|: i=0,1,...,m}, where Sum{d(i)*3^i: i=0,1,...,m} is base 3 representation of n.

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A037836 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.

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A037837 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*5^i: i=0,1,...,m} is the base 5 representation of n.

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A037838 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*6^i: i=0,1,...,m} is the base 6 representation of n.

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Maple

Mathematica

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A037839 a(n) = Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is the base 7 representation of n.

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Mathematica

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A037840 a(n)=Sum{|d(i)-d(i-1)|: i=1,...,m}, where Sum{d(i)*8^i: i=0,1,...,m} is the base 8 representation of n.

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