A094958 -id:A094958 - cp's OEIS frontend

A029744 Numbers of the form 2^n or 3*2^n.

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304

Offset: 1

N. J. A. Sloane

This entry is a list, and so has offset 1. WARNING: However, in this entry several comments, formulas and programs seem to refer to the original version of this sequence which had offset 0. - M. F. Hasler, Oct 06 2014
Number of necklaces with n-1 beads and two colors that are the same when turned over and hence have reflection symmetry. [edited by Herbert Kociemba, Nov 24 2016]
The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - Ralf Stephan, Jun 02 2003
Let k = any nonnegative integer and j = 0 or 1. Then n+1 = 2k + 3j and a(n) = 2^k*3^j. - Andras Erszegi (erszegi.andras(AT)chello.hu), Jul 30 2005
Smallest number having no fewer prime factors than any predecessor, a(0)=1; A110654(n) = A001222(a(n)); complement of A116451. - Reinhard Zumkeller, Feb 16 2006
A093873(a(n)) = 1. - Reinhard Zumkeller, Oct 13 2006
a(n) = a(n-1) + a(n-2) - gcd(a(n-1), a(n-2)), n >= 3, a(1)=2, a(2)=3. - Ctibor O. Zizka, Jun 06 2009
Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). - Reinhard Zumkeller, Aug 08 2011
A002348(a(n)) = A000079(n-3) for n > 2. - Reinhard Zumkeller, Mar 18 2012
Without initial 1, third row in array A228405. - Richard R. Forberg, Sep 06 2013
Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014
Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - Peter Luschny, Oct 30 2019
For n > 1, squares of the terms can be expressed as the sum of two powers of two: 2^x + 2^y. - Karl-Heinz Hofmann, Sep 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012
Index entries for linear recurrences with constant coefficients, signature (0,2).
Index entries for sequences related to necklaces

Cf. A056493, A038754, A063759. Union of A000079 and A007283.
First differences are in A016116(n-1).
Cf. A082125, A094958, A048739, A048985, A193652, A048673, A064216, A246360, A354785.
Row sums of the triangle in sequence A119963. - John P. McSorley, Aug 31 2010
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent. There may be minor differences from (s(n)) at the start, and a shift of indices. A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A060482 (s(n)-3); A136252 (s(n)-3); A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4);   A354785 (3*s(n)), A061776 (3*s(n)-6); A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

a029744 n = a029744_list !! (n-1)
a029744_list = 1 : iterate
   (\x -> if x `mod` 3 == 0 then 4 * x `div` 3 else 3 * x `div` 2) 2
-- Reinhard Zumkeller, Mar 18 2012

1,seq(op([2^i,3*2^(i-1)]),i=1..100); # Robert Israel, Sep 23 2014

CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Function[w, DeleteCases[Union@ Flatten@ w, k_ /; k > Max@ First@ w]]@ TensorProduct[{1, 3}, 2^Range[0, 22]] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{0,2},{1,2,3},50] (* Harvey P. Dale, Jul 04 2017 *)

PARI
```
a(n)=if(n%2,3/2,2)<<((n-1)\2)\1
```

def A029744(n):
    if n == 1: return 1
    elif n % 2 == 0: return 2**(n//2)
    else: return 3 * 2**((n-3)//2) # Karl-Heinz Hofmann, Sep 08 2022

(define (A029744 n) (cond ((<= n 1) n) ((even? n) (expt 2 (/ n 2))) (else (* 3 (expt 2 (/ (- n 3) 2)))))) ;; Antti Karttunen, Sep 23 2014

a(n) = 2*A000029(n) - A000031(n).
For n > 2, a(n) = 2*a(n - 2); for n > 3, a(n) = a(n - 1)*a(n - 2)/a(n - 3). G.f.: (1 + x)^2/(1 - 2*x^2). - Henry Bottomley, Jul 15 2001, corrected May 04 2007
a(0)=1, a(1)=1 and a(n) = a(n-2) * ( floor(a(n-1)/a(n-2)) + 1 ). - Benoit Cloitre, Aug 13 2002
(3/4 + sqrt(1/2))*sqrt(2)^n + (3/4 - sqrt(1/2))*(-sqrt(2))^n. a(0)=1, a(2n) = a(n-1)*a(n), a(2n+1) = a(n) + 2^floor((n-1)/2). - Ralf Stephan, Apr 16 2003 [Seems to refer to the original version with offset=0. - M. F. Hasler, Oct 06 2014]
Binomial transform is A048739. - Paul Barry, Apr 23 2004
E.g.f.: (cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2.
a(1) = 1; a(n+1) = a(n) + A000010(a(n)). - Stefan Steinerberger, Dec 20 2007
u(2)=1, v(2)=1, u(n)=2*v(n-1), v(n)=u(n-1), a(n)=u(n)+v(n). - Jaume Oliver Lafont, May 21 2008
For n => 3, a(n) = sqrt(2*a(n-1)^2 + (-2)^(n-3)). - Richard R. Forberg, Aug 20 2013
a(n) = A064216(A246360(n)). - Antti Karttunen, Sep 23 2014
a(n) = sqrt((17 - (-1)^n)*2^(n-4)) for n >= 2. - Anton Zakharov, Jul 24 2016
Sum_{n>=1} 1/a(n) = 8/3. - Amiram Eldar, Nov 12 2020
a(n) = 2^(n/2) if n is even. a(n) = 3 * 2^((n-3)/2) if n is odd and for n>1. - Karl-Heinz Hofmann, Sep 08 2022

Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000

A003592 Numbers of the form 2^i*5^j with i, j >= 0.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1024, 1250, 1280, 1600, 2000, 2048, 2500, 2560, 3125, 3200, 4000, 4096, 5000, 5120, 6250, 6400, 8000, 8192, 10000, 10240, 12500, 12800

Offset: 1

N. J. A. Sloane

These are the natural numbers whose reciprocals are terminating decimals. - David Wasserman, Feb 26 2002
A132726(a(n), k) = 0 for k <= a(n); A051626(a(n)) = 0; A132740(a(n)) = 1; A132741(a(n)) = a(n). - Reinhard Zumkeller, Aug 27 2007
Where record values greater than 1 occur in A165706: A165707(n) = A165706(a(n)). - Reinhard Zumkeller, Sep 26 2009
Also numbers that are divisible by neither 10k - 7, 10k - 3, 10k - 1 nor 10k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010
A204455(5*a(n)) = 5, and only for these numbers. - Wolfdieter Lang, Feb 04 2012
Since p = 2 and q = 5 are coprime, sum_{n >= 1} 1/a(n) = sum_{i >= 0} sum_{j >= 0} 1/p^i * 1/q^j = sum_{i >= 0} 1/p^i q/(q - 1) = p*q/((p-1)*(q-1)) = 2*5/(1*4) = 2.5. - Franklin T. Adams-Watters, Jul 07 2014
Conjecture: Each positive integer n not among 1, 4 and 12 can be written as a sum of finitely many numbers of the form 2^a*5^b + 1 (a,b >= 0) with no one dividing another. This has been verified for n <= 3700. - Zhi-Wei Sun, Apr 18 2023
1,2 and 4,5 are the only consecutive terms in the sequence. - Robin Jones, May 03 2025

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 73.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (200000 terms)
Math Stack Exchange, Are (1,2) and (4,5) the only two consecutive pairs in A003592, the integers of the form 2^i5^j?
Eric Weisstein's World of Mathematics, Regular Number
Eric Weisstein's World of Mathematics, Decimal Expansion

Complement of A085837. Cf. A094958, A022333 (list of j), A022332 (list of i).
Cf. A003586, A003591, A003593, A003594, A003595, A257997.
Cf. A164768 (difference between consecutive terms)

Filtered([1..10000],n->PowerMod(10,n,n)=0); # Muniru A Asiru, Mar 19 2019

import Data.Set (singleton, deleteFindMin, insert)
a003592 n = a003592_list !! (n-1)
a003592_list = f $ singleton 1 where
   f s = y : f (insert (2 * y) $ insert (5 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

[n: n in [1..10000] | PrimeDivisors(n) subset [2,5]]; // Bruno Berselli, Sep 24 2012

isA003592 := proc(n)
      if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {2,5} = {} );
    end if;
end proc:
A003592 := proc(n)
     option remember;
     if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA003592(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 16 2012

twoFiveableQ[n_] := PowerMod[10, n, n] == 0; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Jan 12 2012 *)
twoFiveableQ[n_] := Union[ MemberQ[{1, 3, 7, 9}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 10]] == {False}; twoFiveableQ[1] = True; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Oct 26 2010 *)
maxExpo = 14; Sort@ Flatten@ Table[2^i * 5^j, {i, 0, maxExpo}, {j, 0, Log[5, 2^(maxExpo - i)]}] (* Or *)
Union@ Flatten@ NestList[{2#, 4#, 5#} &, 1, 7] (* Robert G. Wilson v, Apr 16 2011 *)

list(lim)=my(v=List(),N);for(n=0,log(lim+.5)\log(5),N=5^n;while(N<=lim,listput(v,N);N<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

# A003592.py
from heapq import heappush, heappop
def A003592():
    pq = [1]
    seen = set(pq)
    while True:
        value = heappop(pq)
        yield value
        seen.remove(value)
        for x in 2*value, 5*value:
            if x not in seen:
                heappush(pq, x)
                seen.add(x)
sequence = A003592()
A003592_list = [next(sequence) for _ in range(100)]

from sympy import integer_log
def A003592(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//5**i).bit_length() for i in range(integer_log(x,5)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

def isA003592(n) :
    return not any(d != 2 and d != 5 for d in prime_divisors(n))
@CachedFunction
def A003592(n) :
    if n == 1 : return 1
    k = A003592(n-1) + 1
    while not isA003592(k) : k += 1
    return k
[A003592(n) for n in (1..48)]  # Peter Luschny, Jul 20 2012

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(10*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019
a(n) ~ exp(sqrt(2*log(2)*log(5)*n)) / sqrt(10). - Vaclav Kotesovec, Sep 22 2020
a(n) = 2^A022332(n) * 5^A022333(n). - R. J. Mathar, Jul 06 2025

Incomplete Python program removed by David Radcliffe, Jun 27 2016

A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane, Ralph Peterson (ralphp(AT)library.nrl.navy.mil)

All numbers not equal to some 2^k or 5*2^k [Fraser and Gordon]. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006

T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

T. D. Noe, Table of n, a(n) for n = 1..1000
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

Complement of A094958. Cf. A169580, A000378, A000419, A000408.

For primitive solutions see A005818.

z=100;lst={};Do[a2=a^2;Do[b2=b^2;Do[c2=c^2;e2=a2+b2+c2;e=Sqrt[e2];If[IntegerQ[e]&&e<=z,AppendTo[lst,e]],{c,b,1,-1}],{b,a,1,-1}],{a,1,z}];Union@lst (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)

is(n)=if(n%5,n,n/5)==2^valuation(n,2) \\ Charles R Greathouse IV, Mar 12 2013

def A005767(n):
    def f(x): return n+x.bit_length()+(x//5).bit_length()
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

a(n) = n + 2*log_2(n) + O(1). - Charles R Greathouse IV, Sep 01 2015

A169580(n) = a(n)^2. - R. J. Mathar, Aug 15 2023

More terms from T. D. Noe, Mar 04 2010

A181786 Number of inequivalent solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 3, 0, 2, 3, 3, 0, 6, 2, 3, 1, 2, 1, 8, 1, 3, 3, 4, 0, 10, 2, 5, 3, 4, 3, 8, 0, 5, 6, 6, 2, 11, 3, 6, 1, 8, 2, 12, 1, 6, 8, 8, 1, 15, 3, 8, 3, 7, 4, 20, 0, 6, 10, 9, 2, 16, 5, 9, 3, 9, 4, 15, 3, 15, 8, 10, 0, 22, 5, 11, 6, 9, 6, 18, 2, 11, 11, 14, 3, 21, 6, 13, 1, 12, 8, 31, 2

Offset: 0

T. D. Noe, Nov 12 2010

nonn

Note that a(n)=0 for n=0 and the n in A094958.
Also note that a(2n)=a(n), e.g., a(1000)=a(500)=a(250)=a(125)=14. - Zak Seidov, Mar 02 2012
a(n) is the number of distinct parallelepipeds each one having integer diagonal n and integer sides. - César Eliud Lozada, Oct 26 2014

Samuel Harkness, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Zak Seidov)

Cf. A016725, A016727, A046080, A181787, A181788

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,nn}, {b,a,nn}, {c,b,nn}]; Prepend[t,0]

A164682 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 8.

Original entry on oeis.org

5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304

Offset: 1

Klaus Brockhaus, Aug 21 2009

Interleaving of A020714 and A000079 without initial terms 1, 2, 4.
First differences are in A162255.
Binomial transform is A135532 without initial terms -1, 3. Fourth binomial transform is A164537.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Equals A094958 (numbers of the form 2^n or 5*2^n) without initial terms 1, 2, 4.

Cf. A020714 (5*2^n), A000079 (powers of 2), A162255, A135532, A164537.

[ n le 2 select 2+3*n else 2*Self(n-2): n in [1..40] ];

LinearRecurrence[{0,2},{5,8},60] (* Harvey P. Dale, Jul 20 2022 *)

a(n) = (9-(-1)^n)*2^(1/4*(2*n-5+(-1)^n)).

G.f.: x*(5+8*x)/(1-2*x^2).

A181787 Number of solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 3, 6, 0, 12, 0, 9, 3, 6, 6, 15, 0, 9, 12, 15, 0, 33, 9, 18, 3, 12, 6, 39, 6, 18, 15, 24, 0, 48, 9, 30, 12, 24, 15, 45, 0, 27, 33, 33, 9, 60, 18, 36, 3, 48, 12, 60, 6, 36, 39, 45, 6, 78, 18, 45, 15, 42, 24, 114, 0, 36, 48, 51, 9, 93, 30, 54, 12, 51, 24, 87, 15, 87, 45, 60, 0, 120, 27, 63, 33, 51, 33, 105, 9, 63, 60, 84, 18, 123, 36, 75, 3, 69, 48, 165, 12

Offset: 0

T. D. Noe, Nov 12 2010

Note that a(n)=0 for n=0 and the n in A094958.

			a(3)=3 because 3^2 = 1^2+2^2+2^2 = 2^2+1^2+2^2 = 2^2+2^2+1^2. - _Robert Israel_, Aug 02 2019

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A016725, A016727, A181786, A181788.

N:= 200: # for a(0)..a(N)
A:= Array(0..N):
mults:= [1,3,6]:
for a from 1 while 3*a^2 <= N^2 do
  if a::odd then b0:= a+1; db:= 2 else b0:= a; db:= 1 fi;
  for b from b0 by db while a^2 + 2*b^2 <= N^2 do
    if (a+b)::odd then c0:= b + (b mod 2); dc:= 2 else c0:= b; dc:= 1 fi;
    for c from c0 by dc do
      v:= a^2 + b^2 + c^2;
      if v > N^2 then break fi;
      if issqr(v) then
        w:= sqrt(v);
        A[w]:= A[w]+ mults[nops({a,b,c})];
      fi
od od od:
convert(A,list); # Robert Israel, Aug 02 2019

nn=100; t=Table[0,{nn}]; Do[n=Sqrt[a^2+b^2+c^2]; If[n<=nn && IntegerQ[n], t[[n]]++], {a,nn}, {b,nn}, {c,nn}]; Prepend[t,0]

a(n) = A063691(n^2). - Michel Marcus, Apr 25 2015

a(2*n) = a(n). - Robert Israel, Aug 02 2019

A164053 Partial sums of A162255.

Original entry on oeis.org

3, 5, 11, 15, 27, 35, 59, 75, 123, 155, 251, 315, 507, 635, 1019, 1275, 2043, 2555, 4091, 5115, 8187, 10235, 16379, 20475, 32763, 40955, 65531, 81915, 131067, 163835, 262139, 327675, 524283, 655355, 1048571, 1310715, 2097147, 2621435, 4194299

Offset: 1

Klaus Brockhaus, Aug 08 2009

nonn

Apparently a(n) = A094958(n+4)-5.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 2, -2).

Cf. A162255, A094958.

T:=[ n le 2 select 4-n else 2*Self(n-2): n in [1..39] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

Accumulate[LinearRecurrence[{0,2},{3,2},50]] (* or *) LinearRecurrence[ {1,2,-2},{3,5,11},50] (* Harvey P. Dale, Aug 28 2012 *)

x='x+O('x^50); Vec(x*(3+2*x)/(1-x-2*x^2+2*x^3)) \\ G. C. Greubel, Sep 09 2017

a(n) = 2*a(n-2) + 5 for n > 2; a(1) = 3, a(2) = 5.
a(n) = (13 - 3*(-1)^n)*2^(1/4*(2*n -1 +(-1)^n))/2 - 5.
G.f.: x*(3+2*x)/(1-x-2*x^2+2*x^3).
a(1)=3, a(2)=5, a(3)=11, a(n)=a(n-1)+2*a(n-2)-2*a(n-3). - Harvey P. Dale, Aug 28 2012

A292895 a(n) is the least positive k such that the Hamming weight of k equals the Hamming weight of k + n.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 2, 1, 8, 3, 10, 6, 4, 5, 2, 1, 16, 3, 6, 5, 20, 3, 12, 10, 8, 9, 10, 6, 4, 5, 2, 1, 32, 3, 6, 5, 12, 3, 10, 9, 40, 11, 6, 5, 24, 3, 20, 18, 16, 7, 18, 17, 20, 12, 12, 10, 8, 9, 10, 6, 4, 5, 2, 1, 64, 3, 6, 5, 12, 3, 10, 9, 24, 11, 6, 5, 20, 3, 18, 17, 80, 7, 22, 14, 12, 13, 10, 9, 48

Offset: 0

Rémy Sigrist and Altug Alkan, Sep 26 2017

Inspired by A292849.
The Hamming weight of a number n is given by A000120(n).
Let b(n) be the smallest t such that a(t) = n. Initial values of b(n) are 0, 2, 9, 4, 5, 11, 49, 8, 25, 10, 41, 22, 85, 83, 225, 16, 51, 47, 177, 20, ... See the logarithmic line graph of first 10^3 terms of b(n) sequence in Links section.
Apparently, n = a(n) iff n belongs to A094958. - Rémy Sigrist, Oct 02 2017

			a(49) = 7 since A000120(7) = A000120(7 + 49) and 7 is the least number with this property.

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Altug Alkan, A logarithmic scatterplot of a(n) for n <= 10^6
Altug Alkan, A logarithmic line graph of b(n) for n <= 10^3

Cf. A000120, A094958, A292849.

N:= 1000: # to get all terms before the first where n+a(n)>N
H:= Array(0..N, t -> convert(convert(t,base,2),`+`)):
f:= proc(n) local k;
  for k from 1 to N-n do
    if H[k]=H[k+n] then return k fi
  od:
0
end proc:
R:= NULL:
for n from 0 do
v:= f(n);
if v = 0 then break fi;
  R:= R, v;
od:
R; # Robert Israel, Sep 27 2017

h[n_] := First@ DigitCount[n, 2]; a[n_] := Block[{k=1}, While[h[k] != h[k + n], k++]; k]; Array[a, 90] (* Giovanni Resta, Sep 28 2017 *)

a(n) = {my(k=1); while ((hammingweight(k)) != hammingweight(n+k), k++); k; }

a(n) <= n for n >= 1.
a(2*n) = 2*a(n) for n >= 1.
a(2^m) = 2^m and a(5*2^m) = 5*2^m for m >= 0.
a(2^m - 1) = 1 for m >= 0.
a(2^m + 1) = 3 and a(2^m - 3) = 5 for m >= 3.
a(2^m + 3) = 5 for m >= 4.
a((2^m - 1)^2) = 2^m - 1 for m >= 1.
a(2^(m + 2) + 2^m - 1) = 2^m + 1 m >= 1.
a((2^m + 1)^2) = 7 for m >= 3.

A298359 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, Sum_{k = 1..n} 10^(k-1) * a(k) can be computed without carry in decimal base.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 60, 61, 62, 63, 70, 71, 72, 80, 81, 90, 100, 19, 37, 46, 55, 64, 73, 82, 91, 110, 28, 56, 74

Offset: 1

Rémy Sigrist, Jan 17 2018

More informally: write the terms in decimal under each other, right-justified; the digits on each diagonal in downwards direction sum at most to 9.
The corresponding sequence for base 2 is A094958.
See also A298425 for a similar sequence.

			The first terms, alongside 10^(n-1) * a(n), are:
  n   a(n)  10^(n-1) * a(n)
  --  ----  -------------------
   1   1                      1
   2   2                     20
   3   3                    300
   4   4                   4000
   5   5                  50000
   6   6                 600000
   7   7                7000000
   8   8               80000000
   9   9              900000000
  10  10            10000000000
  11  11           110000000000
  12  12          1200000000000
  13  13         13000000000000
  14  14        140000000000000
  15  15       1500000000000000
  16  16      16000000000000000
  17  17     170000000000000000
  18  18    1800000000000000000
  19  20   20000000000000000000
  20  21  210000000000000000000
The terms on the third column can be summed without carry in decimal base.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A298359

Cf. A094958, A298425.

PARI
```
See Links section.
```

A029745 Expansion of (1 + 2x + 6x^2 + x^3)/(1 - 2x^2).

Original entry on oeis.org

1, 2, 8, 5, 16, 10, 32, 20, 64, 40, 128, 80, 256, 160, 512, 320, 1024, 640, 2048, 1280, 4096, 2560, 8192, 5120, 16384, 10240, 32768, 20480, 65536, 40960, 131072, 81920, 262144, 163840, 524288, 327680, 1048576, 655360, 2097152, 1310720, 4194304

Offset: 1

N. J. A. Sloane

Note that 4 is the only power of 2 not here. All terms are either 2^k or 5*2^k.

Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A094958 (numbers of the form 2^k or 5*2^k).

LinearRecurrence[{0,2},{1,2,8,5},50] (* or *) With[{nn=20},Join[{1,2}, Riffle[ 8*2^Range[0,nn],5 2^Range[0,nn]]]] (* Harvey P. Dale, Sep 28 2016 *)

a(n)=if(n<2,1+max(-1,n),2^(n\2)*if(n%2,5/2,4))

G.f.: (1 + 2x + 6x^2 + x^3)/(1 - 2x^2).

Sum_{n>=1} 1/a(n) = 43/20. - Amiram Eldar, Jan 21 2022

Edited by T. D. Noe, Nov 12 2010

cp's OEIS Frontend

A029744 Numbers of the form 2^n or 3*2^n.

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A003592 Numbers of the form 2^i*5^j with i, j >= 0.

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A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

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A181786 Number of inequivalent solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

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A164682 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 8.

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A181787 Number of solutions to n^2 = a^2 + b^2 + c^2 with positive a, b, c.

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