A049501 -id:A049501 - cp's OEIS frontend

A280726 a(n) = A049501(phi(n)).

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

Offset: 1

Indranil Ghosh, Jan 07 2017

nonn

a(n) = Major index (1st definition) of the total numbers <=n and prime to n i.e., phi(n).

			For n = 10, phi(n) = 4 and major index (1st definition) of 4 = 1. So a(n) = 1.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A000010, A049501, A280306, A280531.

a(n) = A049501(A000010(n)).

A280800 a(n) = A049501(cototient(n)).

Original entry on oeis.org

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

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (1st definition) of cototient(n), where cototient(n) = n - phi(n).

			For n = 10, A051953(n) = 6 and major index (1st definition) of 6 is 2. So, a(10) = 2.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049501, A051953, A280726, A280799.

a(n) = A049501(A051953(n)).

A280306 a(n) = A049501(A003418(n)).

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 4, 13, 13, 16, 16, 28, 28, 46, 46, 46, 46, 77, 77, 48, 48, 48, 48, 117, 117, 141, 141, 134, 134, 213, 213, 326, 326, 326, 326, 326, 326, 352, 352, 352, 352, 389, 389, 413, 413, 413, 413, 508, 508

Offset: 0

Indranil Ghosh, Dec 31 2016

This calculates the major index (1st definition) of the LCM of all the numbers from 1 to n.

			For n=10, the LCM of all the numbers from 1 to 10 is 2520 = 100111011000_2, whose major index (1st definition) is 16, so a(10)=16.

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049501, A003418, A279519.

A280531 a(n) = A049501(A000142(n)).

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 11, 16, 16, 25, 30, 64, 39, 83, 111, 139, 139, 165, 205, 242, 283, 320, 336, 474, 440, 395, 637, 655, 886, 842, 1019, 1159, 1159, 1153, 1327, 1366, 1328, 1243, 1487, 1756, 1623, 2362, 2394, 2274, 2487, 2642, 2907, 2843, 3211, 3049, 3736

Offset: 0

Indranil Ghosh, Jan 04 2017

nonn

a(n) is the major index (1st definition) of n!.

			For n=4, A000142(n) = 24 and A049501(24) = 2. So a(n) = 2.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A000142, A049501.

Cf. A280062 (Major index (2nd definition) of n!).

import math
def M(n):
    x=bin(int(n))[2:]
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s
a=lambda n: M(math.factorial(n))

A280817 a(n) = A049501(sigma(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 2, 5, 2, 3, 3, 2, 2, 0, 5, 1, 4, 9, 1, 5, 2, 4, 0, 9, 4, 3, 4, 5, 1, 0, 2, 7, 2, 5, 6, 4, 3, 11, 9, 2, 5, 9, 7, 5, 2, 5, 3, 6, 5, 8, 7, 4, 5, 4, 4, 11, 4, 9, 5, 2, 6, 0, 9, 5, 6, 6, 2, 5, 5, 2, 11, 9, 5, 7, 2, 9, 4, 13, 4, 6, 9, 3, 7, 7, 4, 11, 11, 15, 3, 9, 1, 5, 4, 6, 8, 9, 7, 7, 8, 7, 6, 13, 2, 11, 7, 7, 8

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (1st definition) of the sum of the divisors of n, i.e., sigma(n).

			For n = 10, sigma(n) = 18 and major index (1st definition) of 18 is 5. So, a(10) = 5.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049501, A000203, A280726, A280800.

Table[Total@ Flatten@ Position[Partition[IntegerDigits[DivisorSigma[1, n], 2], 2, 1], {1, 0}], {n, 120}] (* Michael De Vlieger, Jan 10 2017, after Harvey P. Dale at A049501 *)

a(n) = A049501(A000203(n)).

A049502 Major index of n, 2nd definition.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

a(A023758(n)) = 0; a(A101082(n)) > 0. - Reinhard Zumkeller, Jun 17 2015

			83 = 1010011 has 1's followed by 0's in positions 2 and 5 (reading from the right), so a(83)=7.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; cf. p. 89.

Lars Blomberg, Table of n, a(n) for n = 0..10000

Cf. A049501, A037800.

Cf. A023758, A101082.

a049502 = f 0 1 where
   f m i x = if x <= 4
                then m else f (if mod x 4 == 1
                                  then m + i else m) (i + 1) $ div x 2
-- Reinhard Zumkeller, Jun 17 2015

A049502 := proc(n)
    local a,ndgs,p ;
    a := 0 ;
    ndgs := convert(n,base,2) ;
    for p from 1 to nops(ndgs)-1 do
        if op(p,ndgs)- op(p+1,ndgs) = 1 then
            a := a+p ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

Table[Total[Flatten[Position[Partition[Reverse[IntegerDigits[n,2]],2,1],?(#=={1,0}&)]]],{n,0,110}] (* _Harvey P. Dale, Oct 05 2013 *)
Table[Total[SequencePosition[Reverse[IntegerDigits[n,2]],{1,0}][[All,1]]],{n,0,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 26 2020 *)

a(n)=if(n<5, return(0)); sum(i=0,exponent(n)-1, (bittest(n,i) && !bittest(n,i+1))*(i+1)) \\ Charles R Greathouse IV, Jan 30 2023

def m(n):
    x=bin(int(n))[2:][::-1]
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s
for i in range(101):
    print(str(i)+" "+str(m(i))) # Indranil Ghosh, Dec 22 2016

Write n in binary; add positions where there are 1's followed by 0's, counting from right.

More terms from Erich Friedman, Feb 19 2000

A281388 Write n in binary reflected Gray code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

Original entry on oeis.org

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

Offset: 1

Indranil Ghosh, Jan 21 2017

			For n = 11, the binary reflected Gray code for 11 is '1110'. In '1110', the position of '1' followed immediately to the right by '0' counting from left is 3. So, a(11) = 3.
For n = 12, the binary reflected Gray code for 12 is '1010'. In '1010', the positions of '1' followed immediately to the right by '0' counting from left are 1 and 3. So, a(12) = 1 + 3 = 4.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A003188, A014550, A049501.

def g(n):
    return bin(n^(n//2))[2:]
def a(n):
    x=g(n)
    s=0
    for i in range(1, len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s

a(n) = A049501(A003188(n)).

A280843 a(n) = A049502(sigma(n)).

Original entry on oeis.org

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

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (2nd definition) of the sum of the divisors of n, i.e., sigma(n).

			For n = 10, sigma(10) = 18 and major index (2nd definition) of 18 is 2. So, a(10) = 2.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Indranil Ghosh, Python Program to generate the sequence

Cf. A049501, A000203, A280799, A280815.

a(n) = A049502(A000203(n)).

A281552 Write n in the Elias gamma code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 6, 2, 3, 3, 9, 3, 8, 8, 9, 3, 4, 4, 12, 4, 11, 11, 12, 4, 10, 10, 18, 10, 11, 11, 12, 4, 5, 5, 15, 5, 14, 14, 15, 5, 13, 13, 23, 13, 14, 14, 15, 5, 12, 12, 22, 12, 21, 21, 22, 12, 13, 13, 23, 13, 14, 14, 15, 5, 6, 6, 18, 6, 17, 17, 18, 6, 16, 16, 28, 16, 17, 17, 18, 6

Offset: 1

Indranil Ghosh, Jan 24 2017

			For n = 6 , the Elias gamma code for n is '11010'. In '11010', the positions of '1' followed immediately to the right by '0' counting from left are 2 and 4. So, a(6) = 2 + 4 = 6.
For n = 10, the Elias gamma code for n is '1110010'. In '1110010', the positions of '1' followed immediately to the right by '0' counting from left are 3 and 6. So, a(10) = 3 + 6 = 9.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A049501, A171885, A281149, A281388, A281497.

def a(n):
    x= A281149(n)
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s

a(n) = A049501(A171885(n)) for n > = 1.

cp's OEIS Frontend

A280726 a(n) = A049501(phi(n)).

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A280800 a(n) = A049501(cototient(n)).

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A280306 a(n) = A049501(A003418(n)).

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A280531 a(n) = A049501(A000142(n)).

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A280817 a(n) = A049501(sigma(n)).

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Mathematica

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A049502 Major index of n, 2nd definition.

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A281388 Write n in binary reflected Gray code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

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Formula

A280843 a(n) = A049502(sigma(n)).

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