A300707 -id:A300707 - cp's OEIS frontend

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015
a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]
Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - Reinhard Zumkeller, Sep 29 2008
A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009
Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012
a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013
Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]
The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020
For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Eric Weisstein's World of Mathematics, Kronecker Symbol
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
Index entries for "core" sequences
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,1).

Ones complement of A059841.
Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
Cf. A154955 (Mobius transform), A131577 (binomial transform).
Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
Parity of A005811.

a000035 n = n `mod` 2  -- James Spahlinger, Oct 08 2012

a000035_list = cycle [0,1]  -- Reinhard Zumkeller, Jan 06 2012

A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];

PadLeft[{},110,{0,1}] (* Harvey P. Dale, Sep 25 2011 *)

A000035(n):=mod(n,2)$
makelist(A000035(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=n%2;
```

a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* Ralf Stephan, Mar 27 2015 */

def A000035(n): return n & 1 # Chai Wah Wu, May 25 2022

(define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Mar 21 2017

a(n) = (1 - (-1)^n)/2.
a(n) = n mod 2.
a(n) = 1 - a(n-1).
Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001
G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003
a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005
Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012
a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013
a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015
Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015
a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015
Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024

A087003 a(2n) = 0 and a(2n+1) = mu(2n+1); also the sum of Mobius function values computed for terms of 3x+1 trajectory started at n, provided that Collatz conjecture is true.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1

Offset: 1

Labos Elemer, Oct 02 2003

Observe that (these summatory) terms are from {-1,0,1}, so behave like Mobius function values, not like Mertens function values. Moreover, empirically: a(n) deviates from mu(initial-value) = mu(n) only if iv = n is an even squarefree number (i.e., it is from A039956). - This comment, like also the next one, concerns the original Collatz-related definition of this sequence. - Antti Karttunen, Sep 18 2017
From Marc LeBrun, Feb 19 2004: (Start)
Absolute values are the same as those of A091069. First consider the descending parts of Collatz (or 3x+1) trajectories, those that begin with even numbers 2^p k, with k odd. These go 2^p*k, 2^(p-1)*k, ... 2k, k. All but 2k and k are divisible by 4, a (rational) square, hence their mu values are all 0 and so they contribute nothing to the sum.
Then at the end, since mu(2k) = -mu(k), the last two steps cancel each other out. So every descending chain in a trajectory contributes 0. Of course the full trajectory of every even number consists entirely of descending chains, so A087003 is 0 for all even n.
On the other hand, the trajectory of every odd number consists of just that number followed by the trajectory of an even number (which contributes nothing) so A087003 is indeed equal to mu(n) for odd n.
(End)
The sequence is multiplicative; it may be defined as the Dirichlet inverse of the integers modulo 2 (A000035). - Gerard P. Michon, Apr 29 2007
a(n) appears in the second column of A156241 at every second row. - Mats Granvik, Feb 07 2009

Antti Karttunen, Table of n, a(n) for n = 1..10000
G. P. Michon, The Collatz problem.
G. P. Michon, Multiplicative functions.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A008683, A039956, A292273, A099990, A209229.

Cf. A000035 (the Dirichlet inverse), A318657/A318658 (the "Dirichlet Square Root").

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] lf[x_] := Length[fpl[x]] Table[Apply[Plus, Table[MoebiusMu[Part[fpl[w], j]], {j, 1, lf[w]}]], {w, 1, 256}]
Riffle[MoebiusMu[Range[1,121,2]],0] (* Harvey P. Dale, Jan 24 2025 *)

A006370(n) = if(n%2, 3*n+1, n/2); \\ This function from Michael B. Porter, May 29 2010
A087003(n) = { my(s=1); while(n>1, s += moebius(n); n = A006370(n)); (s); }; \\ Antti Karttunen, Sep 14 2017

a(n)={sumdiv(n, d,  my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 04 2018

A087003(n) = ((n%2)*moebius(n)); \\ Antti Karttunen, Sep 01 2018

a(n) = A008683(n) + A292273(n). - Antti Karttunen, Sep 14 2017
Moebius transform of A209229. - Andrew Howroyd, Aug 04 2018
From Jianing Song, Aug 04 2018: (Start)
Multiplicative with a(2^e) = 0, a(p^e) = (-1 + (-1)^e)/2 for odd primes p.
Dirichlet g.f.: 1/((1 - 2^(-s))*zeta(s)).
(End)
From Antti Karttunen, Sep 01 2018: (Start)
a(n) = A000035(n)*A008683(n).
Dirichlet convolution of A318657/A046644 with itself.
(End)
Sum_{n>=1} a(n)/n^2 = A217739 . Sum_{n>=1} a(n)/n^3 = A233091. Sum_{n>=1} a(n)/n^4 = A300707. - R. J. Mathar, Dec 17 2024

a(2n) = 0, a(2n+1) = mu(2n+1) added to the name as the new primary definition by Antti Karttunen, Sep 18 2017

A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 8, 27, 64, 125, 224, 343, 512, 756, 1008, 1331, 1792, 2197, 2752, 3527, 4096, 4913, 6056, 6859, 8064, 9631, 10656, 12167, 14336, 15750, 17584, 20439, 22016, 24389, 28224, 29791, 32768, 37295, 39312, 43343, 48448, 50653, 54880, 61543, 64512, 68921, 77056, 79507

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^3 * Sum_{d|10, d<10, d odd} 1 / d^3 = 10^3 * (1/1^3 + 1/5^3) = 1008.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), A352048 (k=2), this sequence (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A006519, A051000, A300707.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^3, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..50]); # Robert Israel, Apr 03 2023

A352049[n_]:=DivisorSum[n,1/#^3&,#A352049,50] (* Paolo Xausa, Aug 09 2023 *)
a[n_] := DivisorSigma[-3, n/2^IntegerExponent[n, 2]] * n^3 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^3 * sigma(n >> valuation(n, 2), -3) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^3 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^3 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A051000(n) * A006519(n)^3 - A000035(n).

Sum_{k=1..n} a(k) = c * n^4 / 4, where c = 15*zeta(4)/16 = 1.01467803... (A300707). (End)

A016756 a(n) = (2*n+1)^4.

Original entry on oeis.org

1, 81, 625, 2401, 6561, 14641, 28561, 50625, 83521, 130321, 194481, 279841, 390625, 531441, 707281, 923521, 1185921, 1500625, 1874161, 2313441, 2825761, 3418801, 4100625, 4879681, 5764801, 6765201, 7890481, 9150625, 10556001, 12117361, 13845841, 15752961, 17850625

Offset: 0

N. J. A. Sloane

a(n) is the number of ordered pairs of lattice points (vectors in R^2 with integer coordinates) that are in or on a square centered at the origin with side length 2*n. - Geoffrey Critzer, Apr 20 2013

			a(1) = 81 because there are 9 lattice points in or on the 2 x 2 square centered at the origin, so there are 9*9 =81 ordered pairs. - _Geoffrey Critzer_, Apr 20 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A016755, A060187, A154537, A300707.

[(2*n+1)^4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[(2n+1)^4,{n,0,25}]  (* Geoffrey Critzer, Apr 20 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,81,625,2401,6561},30] (* Harvey P. Dale, Mar 24 2020 *)

vector(40, n, n--; (2*n+1)^4) \\ G. C. Greubel, Sep 15 2018

From Wolfdieter Lang, Mar 12 2017: (Start)
G.f.: (1+76*x+230*x^2+76*x^3+x^4)/(1-x)^5; see row n=5 of A060187.
E.g.f.: (1 + 80*x + 232*x^2 + 128*x^3 + 16*x^4)*exp(x); see row n=4 of A154537. (End)
Sum_{n>=0} 1/a(n) = Pi^4/96 (A300707). - Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = (cos(Pi/sqrt(2)) + cosh(Pi/sqrt(2)))/2.
Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(Pi/2)/8. (End)

A300710 Decimal expansion of 17*Pi^8/161280.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^8), whose value is obtained from zeta(8) given by L. Euler in 1735: Sum_{n>=0} (2n+1)^(-s)=(1-2^(-s))*zeta(s).

			1.0001551790252961193029872492957280415665429750613740...

Jason Bard, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A092736, A111003, A300707, A300709.

MATLAB
```
format long; (17/161280)*pi^8
```
Maple
```
evalf((17/161280)*Pi^8, 120);
```

RealDigits[(17/161280)*Pi^8, 10, 120][[1]]

default(realprecision, 120); (17/161280)*Pi^8

Equals 17*A092736/161280. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,8).
Equals DirichletL(4,1,8).
Equals DirichletL(8,1,8).
Equals DirichletL(16,1,8). (End)
Equals 255*Zeta(8)/256. - Jason Bard, Aug 21 2025

A300709 Decimal expansion of Pi^6/960.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 11 2018

Also the sum of the series Sum_{n>=0} (1/(2n+1)^6), whose value is obtained from zeta(6) given by L. Euler in 1735: Sum_{n>=0}(2n+1)^(-s) = (1-2^(-s))*zeta(s).

			1.0014470766409421219064785871379373946533515917510902...

Index entries for transcendental numbers

Cf. A092732, A111003, A300707, A300710.

MATLAB
```
format long; pi^6/960
```
Maple
```
evalf((1/960)*Pi^6, 120)
```
Mathematica
```
RealDigits[Pi^6/960, 10, 120][[1]]
```
PARI
```
default(realprecision, 120); Pi^6/960
```

Equals A092732/960. - Omar E. Pol, Mar 11 2018
From Artur Jasinski, Jun 24 2025: (Start)
Equals DirichletL(2,1,6).
Equals DirichletL(4,1,6).
Equals DirichletL(8,1,6).
Equals DirichletL(16,1,6). (End)

A300731 Decimal expansion of sqrt(Pi^4/96 - 1).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 11 2018

Also the total harmonic distortion (THD) of a triangle wave, see formula (14) in the Blagouchine & Moreau link.

			0.1211529265193047433149738747453528509885975440568532...

I. V. Blagouchine and E. Moreau, Analytic Method for the Computation of the Total Harmonic Distortion by the Cauchy Method of Residues, IEEE Trans. Commun., vol. 59, no. 9, pp. 2478-2491, 2011. PDF file.
Index entries for transcendental numbers

Cf. A092425, A300690, A300707, A300713, A300714, A300727.

MATLAB
```
format long; sqrt(pi^4/96-1)
```
Maple
```
evalf(sqrt((1/96)*Pi^4-1), 120)
```

RealDigits[Sqrt[Pi^4/96 - 1], 10, 120][[1]]

default(realprecision, 120); sqrt(Pi^4/96-1)

A145378 a(n) = Sum_{d|n} sigma(d) - 2Sum_{2c|n} sigma(c) + 4Sum_{4b|n} sigma(b).

Original entry on oeis.org

1, 2, 5, 7, 7, 10, 9, 20, 18, 14, 13, 35, 15, 18, 35, 49, 19, 36, 21, 49, 45, 26, 25, 100, 38, 30, 58, 63, 31, 70, 33, 110, 65, 38, 63, 126, 39, 42, 75, 140, 43, 90, 45, 91, 126, 50, 49, 245, 66, 76, 95, 105, 55, 116, 91, 180, 105, 62, 61, 245, 63, 66, 162, 235, 105, 130, 69

Offset: 1

N. J. A. Sloane, Mar 12 2009

Dirichlet convolution of [1,-2,0,4,0,0,0,...] with A007429.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. See g(n).

Cf. A007429, A300707.

with(numtheory); g:=proc(n) local d,c,b,t0,t1,t2,t3;
t1:=divisors(n);
t0:=add( sigma(d), d in t1);
t2:=0; for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od:
t3:=0; for d in t1 do if d mod 4 = 0 then t3:=t3+sigma(d/4); fi; od:
t0-2*t2+4*t3; end;
[seq(g(n),n=1..100)];
# alternative
read("transforms") : nmax := 100 :
L27 := [seq(i,i=1..nmax) ];
L := [1,-2,0,4,seq(0,i=1..nmax)] ;
DIRICHLET(L27,L) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

a[n_] := Sum[DivisorSigma[1, d] - 2 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2] + 4 Boole[Mod[d, 4] == 0] DivisorSigma[1, d/4], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Apr 04 2020 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 2^(e + 2) - 3*(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2^(f[i,2]+2) - 3*(f[i,2]+1),  (f[i,1]*(f[i,1]^(f[i,2]+1)-1) - (f[i,1]-1)*(f[i,2]+1))/(f[i,1]-1)^2)); } \\ Amiram Eldar, Oct 25 2022

Dirichlet g.f.: (1-2/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
From Amiram Eldar, Oct 25 2022: (Start)
Multiplicative with a(2^e) = 2^(e+2) - 3*(e+1) and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). (End)

A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 5, 7, 7, 15, 9, 15, 18, 21, 13, 35, 15, 27, 35, 31, 19, 54, 21, 49, 45, 39, 25, 75, 38, 45, 58, 63, 31, 105, 33, 63, 65, 57, 63, 126, 39, 63, 75, 105, 43, 135, 45, 91, 126, 75, 49, 155, 66, 114, 95, 105, 55, 174, 91, 135, 105, 93, 61, 245, 63, 99

Offset: 1

Ilya Gutkovskiy, Sep 13 2019

Inverse Moebius transform of A002131.

Dirichlet convolution of A000027 with A001227.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.

nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]

a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ Andrew Howroyd, Sep 13 2019

G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).
G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} A002131(d).
a(n) = Sum_{d|n} d * A001227(n/d).
a(n) = (A007429(n) + A288417(n)) / 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - Amiram Eldar, Oct 23 2022

A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Oct 31 2021

			0.12931985286416790881897546186483602653397481624314396474709910519161011...

Eric Weisstein's World of Mathematics, Dilogarithm
Index entries for zeta function.

Cf. A006752, A072691, A111003, A153071, A175571, A181983, A300707.

RealDigits[Pi^2/12 - Log[2], 10, 100][[1]] (* Amiram Eldar, Nov 30 2021 *)

-sumalt(n=1, (-1)^n*n/(n+1)^2) \\ Charles R Greathouse IV, Nov 01 2021

Pi^2/12-log(2) \\ Charles R Greathouse IV, Nov 01 2021

from scipy.special import zeta
from math import log
int(''.join(n for n in list(str(zeta(2)/2-log(2)))[2:-2]))

int(str(sum((-1)**(n+1)*n/(n+1)**2 for n in range(1,5000000)))[2:-2])

SageMath
```
(pi^2/12-log(2)).n(digits=100)
```

Equals Pi^2/12-log(2).
Equals Sum_{k>=2} (zeta(k)-zeta(k+1))/2^k. - Amiram Eldar, Mar 20 2022
Equals Integral_{x >= 0} x/(1 + exp(x))^2 dx = (1/2) * Integral_{x >= 0} x*(x - 2)*exp(x)/(1 + exp(x))^2 dx . - Peter Bala, Apr 26 2025

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Haskell

Maple

Mathematica

Maxima

PARI

PARI

Python

Scheme

Formula

A087003 a(2n) = 0 and a(2n+1) = mu(2n+1); also the sum of Mobius function values computed for terms of 3x+1 trajectory started at n, provided that Collatz conjecture is true.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A352049 Sum of the cubes of the divisor complements of the odd proper divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A016756 a(n) = (2*n+1)^4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A300710 Decimal expansion of 17*Pi^8/161280.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

Formula

A300709 Decimal expansion of Pi^6/960.

Views

Author

Keywords

Comments

Examples

A145378 a(n) = Sum_{d|n} sigma(d) - 2Sum_{2c|n} sigma(c) + 4Sum_{4b|n} sigma(b).