A091512 -id:A091512 - cp's OEIS frontend

A143157 Partial sums of A091512.

0, 1, 5, 8, 20, 25, 37, 44, 76, 85, 105, 116, 152, 165, 193, 208, 288, 305, 341, 360, 420, 441, 485, 508, 604, 629, 681, 708, 792, 821, 881, 912, 1104, 1137, 1205, 1240, 1348, 1385, 1461, 1500, 1660, 1701, 1785, 1828, 1960, 2005, 2097, 2144, 2384, 2433, 2533, 2584, 2740, 2793, 2901, 2956, 3180

Offset: 0

Gary W. Adamson, Jul 27 2008

			a(4) = 20 = sum of row 4 terms of triangle A143156, (7 + 6 + 4 + 3).
a(4) = 20 = partial sums of first 4 terms of A091512: (1 + 4 + 3 + 12).
a(4) = 20 = Sum_{j=1..4} j*A001511(j) = 1*1 + 2*2 + 3*1 + 4*3.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A001511, A007814, A091512, A143156, A249152, A249153.

{0}~Join~Accumulate@ Array[IntegerExponent[(2 #)^#, 2] &, 56] (* Michael De Vlieger, Sep 29 2019 *)

def A143157(n): return sum(i*(~i&i-1).bit_length() for i in range(2,2*n+1,2))>>1 # Chai Wah Wu, Jul 11 2022

Partial sums of A091512 = Sum_{j>=1} j*A001511(j), where A001511 is the ruler sequence.
Row sums of triangle A143156.
a(n) = A249152(2*n)/2 = A249153(n) / 2. - Antti Karttunen, Oct 25 2014
a(n) = (1/2)*n*(n + 1) + Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Sep 03 2019; Jan 22 2021
G.f. A(x) satisfies: A(x) = 2*A(x^2)*(1 + x) + x/(1 - x)^3. - Ilya Gutkovskiy, Oct 30 2019
a(n) ~ n^2. - Amiram Eldar, Sep 10 2024

a(0) = 0 prepended and more terms computed by Antti Karttunen, Oct 25 2014

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

Original entry on oeis.org

1, 3, 2, 10, 3, 7, 4, 36, 5, 11, 6, 26, 7, 15, 8, 136, 9, 19, 10, 42, 11, 23, 12, 100, 13, 27, 14, 58, 15, 31, 16, 528, 17, 35, 18, 74, 19, 39, 20, 164, 21, 43, 22, 90, 23, 47, 24, 392, 25, 51, 26, 106, 27, 55, 28, 228, 29, 59, 30, 122, 31, 63, 32, 2080, 33, 67, 34, 138, 35

Offset: 1

Johannes W. Meijer, Dec 24 2012

The a(n) appeared in the analysis of A220002, a sequence related to the Catalan numbers.
The first Maple program makes use of a program by Peter Luschny for the calculation of the a(n) values. The second Maple program shows that this sequence has a beautiful internal structure, see the first formula, while the third Maple program makes optimal use of this internal structure for the fast calculation of a(n) values for large n.
The cross references lead to sequences that have the same internal structure as this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.

Cf. A000027 (the natural numbers), A000120 (1's-counting sequence), A000265 (remove 2's from n), A001316 (Gould's sequence), A001511 (the ruler function), A003484 (Hurwitz-Radon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (Thue-Morse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nim-values), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowhere-zero vector fields), A055975 (Gray code related), A059134, A060789, A060819, A065916, A082392, A085296, A086799, A088837, A089265, A090739, A091512, A091519, A096268, A100892, A103391, A105321 (a fractal sequence), A109168 (a continued fraction), A117973, A129760, A151930, A153733, A160467, A162728, A181988, A182241, A191488 (a companion to Gould's sequence), A193365, A220466 (this sequence).

-- Following Ralf Stephan's recurrence:
import Data.List (transpose)
a220466 n = a006519_list !! (n-1)
a220466_list = 1 : concat
   (transpose [zipWith (-) (map (* 4) a220466_list) a006519_list, [2..]])
-- Reinhard Zumkeller, Aug 31 2014

# First Maple program
a := n -> 2^padic[ordp](n, 2)*(n+1)/2 : seq(a(n), n=1..69); # Peter Luschny, Dec 24 2012
# Second Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 4^p*(n-1)  + 2^(p-1)*(1+2^p) od: od: seq(a(n), n=1..nmax);
# Third Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do n:=2^p: n1:=1: while n <= nmax do a(n) := 4^p*(n1-1)+2^(p-1)*(1+2^p): n:=n+2^(p+1): n1:= n1+1: od: od:  seq(a(n), n=1..nmax);

A220466 = Module[{n, p}, p = IntegerExponent[#, 2]; n = (#/2^p + 1)/2; 4^p*(n - 1) + 2^(p - 1)*(1 + 2^p)] &; Array[A220466, 50] (* JungHwan Min, Aug 22 2016 *)

a(n)=if(n%2,n\2+1,4*a(n/2)-2^valuation(n/2,2)) \\ Ralf Stephan, Dec 17 2013

a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).
a(n) = ((n+1)/2)*(A060818(n)/A060818(n-1))
a(n) = (-1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (-1)^(n+1)*2^(4*n-5)*(2*n)!*A060818(n-1) or q(n) = (1/8)*A220002(n-1)*1/(A098597(2*n-1)/A046161(2*n))*1/(A008991(n-1)/A008992(n-1))
Recurrence: a(2n) = 4a(n) - 2^A007814(n), a(2n+1) = n+1. - Ralf Stephan, Dec 17 2013

A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.

Original entry on oeis.org

1, 2, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

A131577 and A011782 are companions, A131577(n) + A011782(n) = 2^n, (and differences each other). - Paul Curtz, Jan 18 2009
A090127 with offset 0: (1, 2, 2, 4, 8, ...) = A(x) / A(x^2), when A(x) = (1 + 2x + 4x^2 + 8x^3 + ...). - Gary W. Adamson, Feb 20 2010
From Wolfdieter Lang, Apr 18 2012: (Start)
a(n) is the order of 3 modulo 2^n. For n=1 and 2 this is obviously 1 and 2, respectively, and for n >= 3 it is 2^(n-2).
For a proof see, e.g., the Graeme McRae link under A068531, the section 'A Different Approach', proposed by Alexander Monnas, the first part, where the result from the expansion of (4-1)^(2^(k-2)) holds only for k >= 3. See also the Charles R Greathouse IV program below where this result has been used.
This means that the cycle generated by 3, taken modulo 2^n, has length a(n), and that 3 is not a primitive root modulo 2^n, if n >= 3 (because Euler's phi(2^n) = 2^(n-1), n >= 1, see A000010).
(End)
Let r(x) = (1 + 2x + 2x^2 + 4x^3 + ...). Then (1 + 2x + 4x^2 + 8x^3 + ...) = (r(x) * r(x^2) * r(x)^4 * r(x^8) * ...). - Gary W. Adamson, Sep 13 2016

			a(1) = 1 since -1 + 3 = 2 is divisible by 2^1;
a(2) = a(3) = 2 since -1 + 9 = 8 is divisible by 4 = 2^2 and also by 8 = 2^3;
a(5) = 8 since -1 + 6561 = 6560 = 32*205 is divisible by 2^5.
From _Wolfdieter Lang_, Apr 18 2012: (Start)
n=3: the order of 3 (mod 8) is a(3)=2 because the cycle generated by 3 is [3, 3^2==1 (mod 8)].
n=5: a(5) = 2^3 = 8 because the cycle generated by 3 is [3^1=3, 3^2=9, 3^3=27, 17, 19, 25, 11, 1] (mod 32).
  The multiplicative group mod 32 is non-cyclic (see A033949(10)) with the additional four cycles  [5, 25, 29, 17, 21, 9, 13, 1], [7, 17, 23, 1], [15, 1], and [31, 1]. This is the cycle structure of the (Abelian) group Z_8 x Z_2 (see one of the cycle graphs shown in the Wikipedia link 'List of small groups' for the order phi(32)=16, given under A192005).
(End)

Index to divisibility sequences

Cf. A068531, A069895, A088660, A090739, A090740, A091512.

Essentially the same as A000079.

t=Table[Part[Flatten[FactorInteger[ -1+3^(n)]], 2], {n, 1, 130}] Table[Min[Flatten[Position[t, j]]], {j, 1, 10}]
Join[{1,2},2^Range[30]] (* or *) Join[{1,2},NestList[2#&,2,30]] (* Harvey P. Dale, Nov 08 2012 *)

a(n)=2^(n+(n<3)-2) \\ Charles R Greathouse IV, Apr 09 2012

def A090129(n): return n if n<3 else 1<Chai Wah Wu, Jul 11 2022

a(n) = 2^(n-2) if n >= 3, 1 for n=1 and 2 for n=2 (see the order comment above).

a(n+2) = A152046(n) + A152046(n+1) = 2*A011782(n). - Paul Curtz, Jan 18 2009

a(11) through a(20) from R. J. Mathar, Aug 08 2008

More terms (powers of 2, see a comment above) from Wolfdieter Lang, Apr 18 2012

A090739 Exponent of 2 in 9^n - 1.

Original entry on oeis.org

3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 8, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 9, 3, 4, 3, 5, 3, 4, 3, 6, 3, 4, 3, 5, 3, 4, 3, 7, 3, 4, 3, 5, 3, 4, 3

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

The exponent of 2 in the factorization of Fibonacci(6n). - T. D. Noe, Mar 14 2014

Records of 3, 4, 5, 6, 7, 8,.. occur at n= 1, 2, 4, 8, 16, 32,... - R. J. Mathar, Jun 28 2025

			For n = 2, we see that -1 + 3^4 = 80 = 2^4 * 5 so a(2) = 4.
For n = 3, we see that -1 + 3^6 = 728 = 2^3 * 7 * 13, so a(3) = 3.

T. D. Noe, Table of n, a(n) for n = 1..1000
T. Lengyel, The order of the Fibonacci and Lucas numbers, Fib. Quart. 33 (1995), 234-239.

Cf. A024101, A069895, A091512, A088660, A090740, A220466.
Cf. A000005, A006519, A120738 (partial sums).
Appears in A161737.

A090739 := proc(n)
    padic[ordp](9^n-1,2) ;
end proc:
seq(A090739(n),n=1..80) ; # R. J. Mathar, Jun 28 2025

Table[Part[Flatten[FactorInteger[ -1+3^(2*n)]], 2], {n, 1, 70}]
Table[IntegerExponent[Fibonacci[n], 2], {n, 6, 600, 6}] (* T. D. Noe, Mar 14 2014 *)

a(n)=valuation(n,2)+3 \\ Charles R Greathouse IV, Mar 14 2014

def A090739(n): return (~n&n-1).bit_length()+3 # Chai Wah Wu, Jul 11 2022

a(n) = A007814(n) + 3.
a((2*n-1)*2^p) = p + 3, p >= 0. - Johannes W. Meijer, Feb 08 2013
a(n) = log_2(A006519(9^n - 1)). - Alonso del Arte, Feb 08 2013
a(n) = 2*tau(4*n)/(tau(4*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Nov 28 2022

More terms from T. D. Noe, Mar 14 2014

A090740 Exponent of 2 in 3^n - 1.

Original entry on oeis.org

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

Offset: 1

Labos Elemer and Ralf Stephan, Jan 19 2004

Also the 2-adic order of Fibonacci(3n) [Lengyel]. - R. J. Mathar, Nov 05 2008

			n=2: 3^2 - 1 = 8 = 2^3 so a(2)=3.

Robert Israel, Table of n, a(n) for n = 1..10000
T. Lengyel, The order of the Fibonacci and Lucas numbers, Fib. Quart. 33 (1995), 234-239.
Diego Marques and Pavel Trojovský, The p-adic order of some fibonomial coefficients, J. Int. Seq. 18 (2015), Article 15.3.1, proposition 7.

Cf. A069895, A091512, A088660, A090739, A001511.

Cf. A007814, A014445, A059841.

seq(padic:-ordp(3^n-1, 2), n=1..100); # Robert Israel, Dec 28 2015

Table[Part[Flatten[FactorInteger[ -1+3^n]], 2], {n, 1, 70}]
IntegerExponent[#,2]&/@(3^Range[110]-1) (* Harvey P. Dale, Jan 28 2017 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+1+(n/2)%2,1)) /* Ralf Stephan, Jan 23 2004 */

a(n)=valuation(fibonacci(3*n),2); \\ Joerg Arndt, Oct 28 2012

a(n)=my(t=valuation(n,2)); if(t,t+2,1) \\ Charles R Greathouse IV, Mar 14 2014

def A090740(n): return (n&-n).bit_length()+int(not n&1) # Chai Wah Wu, Jul 11 2022

a(n) = A007814(n) + A059841(n) + 1.
Multiplicative with a(p^e) = e+2 if p = 2; 1 if p > 2. G.f.: A(x) = 1/(1-x^2) + Sum_{k>=0} x^(2^k)/(1-x^(2^k)). - Vladeta Jovovic, Jan 19 2004
G.f.: Sum_{k>=0} t*(1+2*t+t^2+t^3)/(1-t^4) with t=x^2^k. Recurrence: a(2n) = a(n) + 1 + [n odd], a(2n+1) = 1. - Ralf Stephan, Jan 23 2004
a(n) = A337923(3*n). [Lengyel]. - R. J. Mathar, Nov 05 2008
G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x) + x^2/(1-x^4). - Robert Israel, Dec 28 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2. - Amiram Eldar, Nov 28 2022
Dirichlet g.f.: zeta(s)*(2^s+1-1/2^s)/(2^s-1). - Amiram Eldar, Jan 04 2023

A069895 2^a(n) divides (2n)^(2n): exponent of 2 in (2n)^(2n).

Original entry on oeis.org

2, 8, 6, 24, 10, 24, 14, 64, 18, 40, 22, 72, 26, 56, 30, 160, 34, 72, 38, 120, 42, 88, 46, 192, 50, 104, 54, 168, 58, 120, 62, 384, 66, 136, 70, 216, 74, 152, 78, 320, 82, 168, 86, 264, 90, 184, 94, 480, 98, 200, 102, 312, 106, 216, 110, 448, 114, 232, 118, 360, 122

Offset: 1

Labos Elemer, Apr 10 2002

Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.

Cf. A001511, A007814, A085534, A091512, A249153 (partial sums).

function A069895List(length)
    a = zeros(Int, length)
    for n in 1:length a[n] = 2 * (isodd(n) ? n : n + a[div(n, 2)]) end
a end
A069895List(61) |> println # Peter Luschny, Oct 16 2021

a:= 2*n*padic[ordp](2*n, 2):
seq(a(n), n=1..61);  # Alois P. Heinz, Oct 14 2021

Table[ Part[ Flatten[ FactorInteger[n^n]], 2], {n, 2, 124, 2}]

a(n) = n<<=1; n*valuation(n,2); \\ Kevin Ryde, Oct 14 2021

def A069895(n): return n*(n&-n).bit_length()<<1 # Chai Wah Wu, Jul 11 2022

a(n) = 2*n*A001511(n).
a(n) = A007814(A085534(n)). [corrected by Kevin Ryde, Oct 15 2021]
G.f.: Sum_{k>=0} 2^(k+1)*x^2^k/(1-x^2^k)^2. - Ralf Stephan, Jun 07 2003
a(n) = 2 * A091512(n). - Alois P. Heinz, Oct 14 2021
Sum_{k=1..n} a(k) ~ 2*n^2. - Amiram Eldar, Sep 13 2024

A249152 Exponent of 2 in the hyperfactorials: a(n) = A007814(A002109(n)).

Original entry on oeis.org

0, 0, 2, 2, 10, 10, 16, 16, 40, 40, 50, 50, 74, 74, 88, 88, 152, 152, 170, 170, 210, 210, 232, 232, 304, 304, 330, 330, 386, 386, 416, 416, 576, 576, 610, 610, 682, 682, 720, 720, 840, 840, 882, 882, 970, 970, 1016, 1016, 1208, 1208, 1258, 1258, 1362, 1362, 1416, 1416, 1584, 1584, 1642, 1642

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

This is the function ord_2(D*_n) listed in the leftmost column of Table 7.1 in Lagarias & Mehta 2014 paper (on page 19).

Amiram Eldar, Table of n, a(n) for n = 0..10000
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Luca Onnis, On the p-adic valuation of a hyperfactorial, arXiv:2109.05616 [math.NT], 2021.

Bisection: A249153.
Cf. A174605, A187059, A002109, A007814, A091512, A143157.
Cf. A133457 (binary exponents).

[0] cat [&+[i*Valuation(i, 2):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Oct 18 2019

with(padic): seq(add(i*ordp(i, 2), i=1..n), n=0..60); # Ridouane Oudra, Oct 17 2019

Table[i=0;Hyperfactorial@n//.x_/;EvenQ@x:>(i++;x/2);i,{n,0,60}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

a(n) = sum(i=1, n, i*valuation(i, 2)); \\ Michel Marcus, Sep 14 2021

a(n) = my(v=binary(n),t=0); forstep(j=#v,1,-1, if(v[j],v[j]=t--,t++)); (n^2 + fromdigits(v,2))>>1; \\ Kevin Ryde, Nov 03 2021

def A249152(n): return sum(i*(~i&i-1).bit_length() for i in range(2,n+1,2)) # Chai Wah Wu, Jul 11 2022

a(n) = 2 * A143157(floor(n/2)).
a(n) = A174605(n) + A187059(n). [Lagarias and Mehta theorem 4.1 for p=2]
a(n) = Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Oct 17 2019
a(n) ~ (n^2+2n)/2 as n -> infinity. - Luca Onnis, Oct 17 2021
a(n) ~ ((A011371(n))^2)/2 as n -> infinity. - Luca Onnis, Nov 02 2021
From Kevin Ryde, Nov 03 2021: (Start)
a(2n) = a(2n+1) = 2*a(n) + n*(n+1).
a(n) = ( n^2 + Sum_{j=1..k} (e[j]-2*j+1) * 2^e[j] )/2, where binary expansion n = 2^e[1] + ... + 2^e[k] with ascending exponents e[1] < e[2] < ... < e[k] (A133457).
(End)
a(n) = Sum_{j=1..floor(log_2(n))} j*2^j*round(n/2^(j+1))^2, for n>=1. - Ridouane Oudra, Oct 01 2022

A162728 G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.

Original entry on oeis.org

1, 3, 2, 8, 4, 6, 6, 20, 6, 12, 10, 16, 12, 18, 8, 48, 16, 18, 18, 32, 12, 30, 22, 40, 20, 36, 18, 48, 28, 24, 30, 112, 20, 48, 24, 48, 36, 54, 24, 80, 40, 36, 42, 80, 24, 66, 46, 96, 42, 60, 32, 96, 52, 54, 40, 120, 36, 84, 58, 64, 60, 90, 36, 256, 48, 60, 66, 128, 44, 72, 70

Offset: 1

Paul D. Hanna, Jul 12 2009

Dirichlet inverse of A117212. - R. J. Mathar, Jul 15 2010

			x/(1-x) = log(1+x) + 3*log(1+x^2)/2 + 2*log(1+x^3)/3 + 8*log(1+x^4)/4 + 4*log(1+x^5)/5 + 6*log(1+x^6)/6 + 6*log(1+x^7)/7 + 20*log(1+x^8)/8 +...

Paul D. Hanna, Table of n, a(n) for n = 1..10000

Cf. A090739, A091512, A000010 (Euler phi), A220466.

Cf. A007814, A001511, A090740, A085058, A055034, A008683.

nmax:=71: with(numtheory): for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+2)*2^(p-1)*phi(2*n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 26 2013

f[p_, e_] := (p-1)*p^(e-1); f[2, e_] := (e+2)*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

/* As the inverse Mobius transform of A091512: */
{a(n)=sumdiv(n,d,moebius(n/d)*valuation((2*d)^d,2))}

/* From a(2n-1)=phi(2n-1); a(2n)=phi(2n)*A090739(n), we get: */
{a(n)=if(n%2==1,eulerphi(n),eulerphi(n)*valuation(3^n-1,2))}

/* From x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n, we get: */
{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0);A[ #A]=#A*(1-polcoeff(sum(m=1,#A,A[m]/m*log(1+x^m +x*O(x^#A)) ),#A)));A[n]}

a(2n-1) = phi(2n-1); a(2n) = phi(2n)*A090739(n), where A090739(n) = exponent of 2 in 3^(2n)-1.
Inverse Mobius transform of A091512, where A091512(n) = exponent of 2 in (2n)^n.
Multiplicative: a(m*n) = a(m)*a(n) when gcd(m,n)=1, with a(p) = p-1 for odd prime p and a(2)=3.
G.f.: x/(1-x)^2 = Sum_{n>=1} a(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
Dirichlet g.f.: zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - R. J. Mathar, Apr 14 2011
a((2*n-1)*2^p) = (p+2)*2^(p-1)* phi(2*n-1), p >= 0. Observe that a(2^p) = A001792(p). - Johannes W. Meijer, Jan 26 2013
Sum_{k=1..n} a(k) ~ 6*n^2 / Pi^2. - Vaclav Kotesovec, Feb 07 2019
Multiplicative with a(2^e) = (e+2)*2^(e-1) and a(p^e) = (p-1)*p^(e-1) for an odd prime p. - Amiram Eldar, Aug 27 2023
From Ridouane Oudra, Jul 05 2025: (Start)
a(n) = Sum_{k=0..A007814(n)} 2^k*phi(n/2^k).
a(n) = Sum_{d|n} mu(n/d)*d*A001511(d).
a(n) = A000010(n)*A090740(n).
a(n) = A085058(n-1)*A055034(n), for n>1. (End)

A364222 Expansion of Sum_{k>=0} 3^k * x^(3^k) / (1 - x^(3^k))^2.

Original entry on oeis.org

1, 2, 6, 4, 5, 12, 7, 8, 27, 10, 11, 24, 13, 14, 30, 16, 17, 54, 19, 20, 42, 22, 23, 48, 25, 26, 108, 28, 29, 60, 31, 32, 66, 34, 35, 108, 37, 38, 78, 40, 41, 84, 43, 44, 135, 46, 47, 96, 49, 50, 102, 52, 53, 216, 55, 56, 114, 58, 59, 120, 61, 62, 189, 64, 65, 132, 67, 68, 138, 70, 71, 216, 73, 74

Offset: 1

Seiichi Manyama, Jul 14 2023

Cf. A007949, A051064, A327625.

Cf. A091512, A364223, A364224.

a[n_] := n * (IntegerExponent[n, 3] + 1); Array[a, 100] (* Amiram Eldar, Jul 14 2023 *)

PARI
```
a(n) = n*(valuation(n, 3)+1);
```

a(n) = n * A051064(n).
If n == 0 (mod 3), a(n) = n + 3 * a(n/3) otherwise a(n) = n.
From Amiram Eldar, Jul 14 2023: (Start)
Multiplicative with a(3^e) = (e+1)*3^e and a(p^e) = p*e if p != 3.
Dirichlet g.f.: (3^s/(3^s-3)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4)*n^2. (End)

A249154 (n+1) times the number of 1's in the binary expansion of n.

Original entry on oeis.org

0, 2, 3, 8, 5, 12, 14, 24, 9, 20, 22, 36, 26, 42, 45, 64, 17, 36, 38, 60, 42, 66, 69, 96, 50, 78, 81, 112, 87, 120, 124, 160, 33, 68, 70, 108, 74, 114, 117, 160, 82, 126, 129, 176, 135, 184, 188, 240, 98, 150, 153, 208, 159, 216, 220, 280, 171, 232, 236, 300, 244, 310, 315, 384, 65

Offset: 0

Antti Karttunen, Nov 02 2014

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000120, A000788, A187059, A245788, A091512.

A249154[n_] := (n + 1)*DigitCount[n, 2, 1]; Array[A249154, 100, 0] (* Paolo Xausa, Feb 11 2025 *)

a(n) = (n+1)*hammingweight(n); \\ Michel Marcus, Feb 11 2025

def A249154(n): return (n+1)*n.bit_count() # Chai Wah Wu, Nov 11 2024

(define (A249154 n) (* (+ n 1) (A000120 n)))

a(n) = (n+1) * A000120(n).
a(n) = A000120(n) + A245788(n).
a(n) = 2*A000788(n) - A187059(n).

cp's OEIS Frontend

A143157 Partial sums of A091512.

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A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1.

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A090129 Smallest exponent such that -1 + 3^a(n) is divisible by 2^n.

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A090739 Exponent of 2 in 9^n - 1.

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A090740 Exponent of 2 in 3^n - 1.

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A069895 2^a(n) divides (2n)^(2n): exponent of 2 in (2n)^(2n).

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A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.