A049502 -id:A049502 - cp's OEIS frontend

A280799 a(n) = A049502(phi(n)).

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

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = Major index (2nd definition) of the total numbers <=n and prime to n i.e., phi(n).

			For n = 11, phi(n) = 10 and major index (2nd definition) of 10 = 2. So a(n) = 2.

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

Cf. A000010, A049502, A279519, A280062, A280726.

a(n) = A049502(A000010(n)).

A279519 a(n) = A049502(A003418(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 9, 11, 14, 14, 23, 23, 42, 42, 42, 46, 79, 79, 48, 48, 48, 48, 95, 95, 125, 125, 180, 180, 182, 182, 307, 320, 320, 320, 320, 320, 365, 365, 365, 365, 390, 390, 451, 451, 451, 451, 360, 360, 564, 564, 564, 564, 582, 582, 582, 582, 582, 582, 745, 745, 804, 804, 804, 822, 822, 822, 866

Offset: 0

Indranil Ghosh, Dec 14 2016

			For n=10, the LCM of all the numbers from 1 to 10 is 2520 = 100111011000_2, whose major index (2nd definition) is 14, so a(10)=14.

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

Cf. A003418, A049502.

Map[Total@ SequencePosition[Reverse@ #, {1, 0}][[All, 1]] &@ IntegerDigits[#, 2] &, {1}~Join~Table[LCM @@ Range@ n, {n, 67}]] (* Michael De Vlieger, Dec 16 2016, Version 10.1 *)

A280062 a(n) = A049502(A000142(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 13, 16, 22, 38, 47, 73, 60, 127, 160, 166, 194, 249, 348, 345, 359, 497, 532, 682, 709, 727, 1000, 887, 1312, 1155, 1297, 1934, 2069, 1722, 1796, 2148, 2337, 1839, 2595, 2774, 2440, 3314, 3450, 3253, 3379, 3786, 4466, 4366, 4795, 5189, 5598

Offset: 0

Indranil Ghosh, Jan 04 2017

nonn

a(n) is the major index (2nd definition) of n!.

			for n=15, A000142(n) = 1307674368000 and A049502(1307674368000) = 166. So a(n) = 166.

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

Cf. A049502, A000142.

import math
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
a=lambda n: m(math.factorial(n))

A280815 a(n) = A049502(cototient(n)).

Original entry on oeis.org

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

Offset: 1

Indranil Ghosh, Jan 08 2017

nonn

a(n) = major index (2nd definition) of cototient(n), where cototient(n) = n - phi(n).

			For n = 30, cototient(n) = 22 and major index (2nd definition) of 22 is 3. So, a(n) = 3.

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

Cf. A049502, A051953, A280726, A280799, A280800.

a(n) = A049502(A051953(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)).

A023758 Numbers of the form 2^i - 2^j with i >= j.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 96, 112, 120, 124, 126, 127, 128, 192, 224, 240, 248, 252, 254, 255, 256, 384, 448, 480, 496, 504, 508, 510, 511, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1023

Offset: 1

Olivier Gérard

Numbers whose digits in base 2 are in nonincreasing order.
Might be called "nialpdromes".
Subset of A077436. Proof: Since a(n) is of the form (2^i-1)*2^j, i,j >= 0, a(n)^2 = (2^(2i) - 2^(i+1))*2^(2j) + 2^(2j) where the first sum term has i-1 one bits and its 2j-th bit is zero, while the second sum term switches the 2j-th bit to one, giving i one bits, as in a(n). - Ralf Stephan, Mar 08 2004
Numbers whose binary representation contains no "01". - Benoit Cloitre, May 23 2004
Every polynomial with coefficients equal to 1 for the leading terms and 0 after that, evaluated at 2. For instance a(13) = x^4 + x^3 + x^2 at 2, a(14) = x^4 + x^3 + x^2 + x at 2. - Ben Paul Thurston, Jan 11 2008
From Gary W. Adamson, Jul 18 2008: (Start)
As a triangle by rows starting:
   1;
   2,  3;
   4,  6,  7;
   8, 12, 14, 15;
  16, 24, 28, 30, 31;
  ...,
equals A000012 * A130123 * A000012, where A130123 = (1, 0,2; 0,0,4; 0,0,0,8; ...). Row sums of this triangle = A000337 starting (1, 5, 17, 49, 129, ...). (End)
First differences are A057728 = 1; 1; 1; 1; 2,1; 1; 4,2,1; 1; 8,4,2,1; 1; ... i.e., decreasing powers of 2, separated by another "1". - M. F. Hasler, May 06 2009
Apart from first term, numbers that are powers of 2 or the sum of some consecutive powers of 2. - Omar E. Pol, Feb 14 2013
From Andres Cicuttin, Apr 29 2016: (Start)
Numbers that can be digitally generated with twisted ring (Johnson) counters. This is, the binary digits of a(n) correspond to those stored in a shift register where the input bit of the first bit storage element is the inverted output of the last storage element. After starting with all 0’s, each new state is obtained by rotating the stored bits but inverting at each state transition the last bit that goes to the first position (see link).
Examples: for a(n) represented by three bits
             Binary
a(5)= 4   ->  100   last bit = 0
a(6)= 6   ->  110   first bit = 1 (inverted last bit of previous number)
a(7)= 7   ->  111
and for a(n) represented by four bits
             Binary
a(8) = 8   -> 1000
a(9) = 12  -> 1100  last bit = 0
a(10)= 14  -> 1110  first bit = 1 (inverted last bit of previous number)
a(11)= 15  -> 1111
(End)
Powers of 2 represented in bases which are terms of this sequence must always contain at least one digit which is also a power of 2. This is because 2^i mod (2^i - 2^j) = 2^j, which means the last digit always cycles through powers of 2 (or if i=j+1 then the first digit is a power of 2 and the rest are trailing zeros). The only known non-member of this sequence with this property is 5. - Ely Golden, Sep 05 2017
Numbers k such that k = 2^(1 + A000523(k)) - 2^A007814(k). - Daniel Starodubtsev, Aug 05 2021
A002260(n) = v(a(n)/2^v(a(n))+1) and A002024(n) = A002260(n) + v(a(n)) where v is the dyadic valuation (i.e., A007814). - Lorenzo Sauras Altuzarra, Feb 01 2023

			a(22) = 64 = 32 + 32 = 2^5 + a(16) = 2^A003056(20) + a(22-5-1).
a(23) = 96 = 64 + 32 = 2^6 + a(16) = 2^A003056(21) + a(23-6-1).
a(24) = 112 = 64 + 48 = 2^6 + a(17) = 2^A003056(22) + a(24-6-1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5051 terms from T. D. Noe)
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
S. M. Shabab Hossain, Md. Mahmudur Rahman and M. Sohel Rahman, Solving a Generalized Version of the Exact Cover Problem with a Light-Based Device, Optical Supercomputing, Lecture Notes in Computer Science, 2011, Volume 6748/2011, 23-31, DOI: 10.1007/978-3-642-22494-2_4.
Eric Weisstein's World of Mathematics, Digit.
Wikipedia, Ring counter.
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A000337(r) = sum of row T(r, c) with 0 <= c < r. See also A002024, A003056, A140129, A140130, A221975.
Cf. A007088, A130123, A101082 (complement), A340375 (characteristic function).
This is the base-2 version of A064222. First differences are A057728.
Subsequence of A077436, of A129523, of A277704, and of A333762.
Subsequences: A043569 (nonzero even terms, or equally, nonzero terms doubled), A175332, A272615, A335431, A000396 (its even terms only), A324200.
Positions of zeros in A049502, A265397, A277899, A284264.
Positions of ones in A283983, A283989.
Positions of nonzero terms in A341509 (apart from the initial zero).
Positions of squarefree terms in A260443.
Fixed points of A264977, A277711, A283165, A334666.
Distinct terms in A340632.
Cf. also A002024, A002260, A007814, A133797, A152449, A181666, A211705, A277699, A306441.
Cf. also A309758, A309759, A309761 (for analogous sequences).

import Data.Set (singleton, deleteFindMin, insert)
a023758 n = a023758_list !! (n-1)
a023758_list = 0 : f (singleton 1) where
f s = x : f (if even x then insert z s' else insert z $ insert (z+1) s')
where z = 2*x; (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 24 2014, Dec 19 2012

a:=proc(n) local n2,d: n2:=convert(n,base,2): d:={seq(n2[j]-n2[j-1],j=2..nops(n2))}: if n=0 then 0 elif n=1 then 1 elif d={0,1} or d={0} or d={1} then n else fi end: seq(a(n),n=0..2100); # Emeric Deutsch, Apr 22 2006

Union[Flatten[Table[2^i - 2^j, {i, 0, 100}, {j, 0, i}]]] (* T. D. Noe, Mar 15 2011 *)
Select[Range[0, 2^10], NoneTrue[Differences@ IntegerDigits[#, 2], # > 0 &] &] (* Michael De Vlieger, Sep 05 2017 *)

for(n=0,2500,if(prod(k=1,length(binary(n))-1,component(binary(n),k)+1-component(binary(n),k+1))>0,print1(n,",")))

A023758(n)= my(r=round(sqrt(2*n--))); (1<<(n-r*(r-1)/2)-1)<<(r*(r+1)/2-n)
/* or, to illustrate the "decreasing digit" property and analogy to A064222: */
A023758(n,show=0)={ my(a=0); while(n--, show & print1(a","); a=vecsort(binary(a+1)); a*=vector(#a,j,2^(j-1))~); a} \\ M. F. Hasler, May 06 2009

is(n)=if(n<5,1,n>>=valuation(n,2);n++;n>>valuation(n,2)==1) \\ Charles R Greathouse IV, Jan 04 2016

list(lim)=my(v=List([0]),t); for(i=1,logint(lim\1+1,2), t=2^i-1; while(t<=lim, listput(v,t); t*=2)); Set(v) \\ Charles R Greathouse IV, May 03 2016

def a_next(a_n): return (a_n | (a_n >> 1)) + (a_n & 1)
a_n = 1; a = [0]
for i in range(55): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022

from math import isqrt
def A023758(n): return (1<<(m:=isqrt(n-1<<3)+1>>1))-(1<<(m*(m+1)-(n-1<<1)>>1)) # Chai Wah Wu, Feb 23 2025

a(n) = 2^s(n) - 2^((s(n)^2 + s(n) - 2n)/2) where s(n) = ceiling((-1 + sqrt(1+8n))/2). - Sam Alexander, Jan 08 2005
a(n) = 2^k + a(n-k-1) for 1 < n and k = A003056(n-2). The rows of T(r, c) = 2^r-2^c for 0 <= c < r read from right to left produce this sequence: 1; 2, 3; 4, 6, 7; 8, 12, 14, 15; ... - Frank Ellermann, Dec 06 2001
For n > 0, a(n) mod 2 = A010054(n). - Benoit Cloitre, May 23 2004
A140130(a(n)) = 1 and for n > 1: A140129(a(n)) = A002262(n-2). - Reinhard Zumkeller, May 14 2008
a(n+1) = (2^(n - r(r-1)/2) - 1) 2^(r(r+1)/2 - n), where r=round(sqrt(2n)). - M. F. Hasler, May 06 2009
Start with A000225. If k is in the sequence, then so is 2k. - Ralf Stephan, Aug 16 2013
G.f.: (x^2/((2-x)*(1-x)))*(1 + Sum_{k>=0} x^((k^2+k)/2)*(1 + x*(2^k-1))). The sum is related to Jacobi theta functions. - Robert Israel, Feb 24 2015
A049502(a(n)) = 0. - Reinhard Zumkeller, Jun 17 2015
a(n) = a(n-1) + a(n-d)/a(d*(d+1)/2 + 2) if n > 1, d > 0, where d = A002262(n-2). - Yuchun Ji, May 11 2020
A277699(a(n)) = a(n)^2, A306441(a(n)) = a(n+1). - Antti Karttunen, Feb 15 2021 (the latter identity from A306441)
Sum_{n>=2} 1/a(n) = A211705. - Amiram Eldar, Feb 20 2022

Definition changed by N. J. A. Sloane, Jan 05 2008

A101082 Numbers n such that binary representation contains bit strings "10" and "01" (possibly overlapping).

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Rick L. Shepherd, Nov 29 2004

Subsequence of A062289; set difference A062289 minus A043569.
Complement of A023758. Also numbers not the sum of consecutive powers of 2. - Omar E. Pol, Mar 04 2013
Equivalently, numbers not the difference of two powers of two. - Charles R Greathouse IV, Mar 07 2013
The terms >=9 are bases in which a power of 2 exists, which does not contain a digit that is a power of 2. In base 10, 2^16 = 65536 is such a number, as it does not contain any one-digit power of 2, which in base 10 are 1, 2, 4 and 8. - Patrick Wienhöft, Jul 28 2016

			29 = 11101_2 is a term, "10" and "01" are contained (here overlapping).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Patrick Wienhöft, Python program
Index entries for 2-automatic sequences.

Complement: A023758.

Cf. A043569, A062289, A049502.

a101082 n = a101082_list !! (n-1)
a101082_list = filter ((> 0) . a049502) [0..]
-- Reinhard Zumkeller, Jun 17 2015

Select[Range@ 120, Function[d, Times @@ Total@ Map[Map[Function[k, Boole@ MatchQ[#, k]], {{1, 0}, {0, 1}}] &, Partition[d, 2, 1]] > 0]@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Dec 23 2016 *)
Select[Range[100],With[{c=IntegerDigits[#,2]},SequenceCount[c,{1,0}]>0&&SequenceCount[c,{0,1}]>0]&] (* Harvey P. Dale, Jun 08 2025 *)

is(n)=n>>=valuation(n,2);n+1!=1<Charles R Greathouse IV, Mar 07 2013

def A101082(n):
    def f(x): return n+((k:=x.bit_length())*(k-1)>>1)+sum(1 for i in range(k) if (1<Chai Wah Wu, Feb 23 2025

a(n) ~ n. In particular a(n) = n + (log_2 n)^2/2 + O(log n). - Charles R Greathouse IV, Mar 07 2013

A049502(a(n)) > 0. - Reinhard Zumkeller, Jun 17 2015

A049501 Major index of n, first definition.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			50 = 110010 has 1's followed by 0's in positions 2 and 5 (reading from the left), so a(50)=7. At the beginning of the sequence we have 0->0, 1->0, 10->1, 11->0, 100->1, 101->1, 110->2, 111->0, 1000->1, 1001->1, 1010->1+3=4, ...

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

J. Stauduhar, Table of n, a(n) for n = 0..10000

Cf. A049502.

a[n_] := Total[ Flatten[ Position[ IntegerDigits[n, 2] //. {b___, 1, 0, c___} -> {b, 2, 3, c}, 2]]]; Table[a[n], {n, 0, 102}] (* Jean-François Alcover, Dec 20 2011 *)
Table[Total[Flatten[Position[Partition[IntegerDigits[n,2],2,1],{1,0}]]],{n,0,110}] (* Harvey P. Dale, Nov 04 2012 *)
Table[Total[SequencePosition[IntegerDigits[n,2],{1,0}][[;;,1]]],{n,0,110}] (* Harvey P. Dale, Feb 19 2023 *)

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

More terms from Erich Friedman, Feb 19 2000

A281497 Write n in binary reflected Gray code and sum the positions where there is a '1' followed immediately to the left by a '0', counting the rightmost digit as position 1.

Original entry on oeis.org

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

Offset: 1

Indranil Ghosh, Jan 23 2017

			For n = 12, the binary reflected Gray code for 12 is '1010'. In '1010', the position of '1' followed immediately to the left by a '0' counting from right is 2. So, a(12) = 2.

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

Cf. A003188, A014550, A049502, A281388.

Table[If[Length@ # == 0, 0, Total[#[[All, 1]]]] &@ SequencePosition[ Reverse@ IntegerDigits[#, 2] &@ BitXor[n, Floor[n/2]], {1, 0}], {n, 120}] (* Michael De Vlieger, Jan 23 2017, Version 10.1, after Robert G. Wilson v at A003188 *)

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

a(n) = A049502(A003188(n)).

cp's OEIS Frontend

A280799 a(n) = A049502(phi(n)).

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A279519 a(n) = A049502(A003418(n)).

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Mathematica

A280062 a(n) = A049502(A000142(n)).

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A280815 a(n) = A049502(cototient(n)).

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

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A023758 Numbers of the form 2^i - 2^j with i >= j.

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Maple

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A101082 Numbers n such that binary representation contains bit strings "10" and "01" (possibly overlapping).

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