A023758 - OEIS

OFFSET

1,3

COMMENTS

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

LINKS

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.

FORMULA

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

EXAMPLE

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).

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) 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, ", ")))

(PARI) 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

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

(PARI) 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

(Haskell)

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

(Python)

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

CROSSREFS

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).

Sequence in context: A077436 A277704 A082752 * A054784 A018585 A018399

Adjacent sequences: A023755 A023756 A023757 * A023759 A023760 A023761

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

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

STATUS

approved