A353908 -id:A353908 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A175254 a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.

Original entry on oeis.org

1, 5, 13, 28, 49, 82, 123, 179, 248, 335, 434, 561, 702, 867, 1056, 1276, 1514, 1791, 2088, 2427, 2798, 3205, 3636, 4127, 4649, 5213, 5817, 6477, 7167, 7929, 8723, 9580, 10485, 11444, 12451, 13549, 14685, 15881, 17133, 18475, 19859, 21339, 22863, 24471, 26157

Offset: 1

Jaroslav Krizek, Mar 14 2010

Partial sums of A024916. - Omar E. Pol, Jul 03 2014
a(n) is also the volume of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Aug 12 2015
Also the alternating row sums of A262612. - Omar E. Pol, Nov 23 2015
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and A000027.
Convolution of A340793 and the nonzero terms of A000217.
Antidiagonal sums of A319073.
Row sums of A274824. (End)
Row sums of A345272. - Omar E. Pol, Jun 14 2021
Also the alternating row sums of A353690. - Omar E. Pol, Jun 05 2022

			For n = 4: a(4) = sigma(1)*4 + sigma(2)*3 + sigma(3)*2 + sigma(4)*1 = 1*4 + 3*3 + 4*2 + 7*1 = 28.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2209 terms from Indranil Ghosh)

Cf. A000203, A000217, A006218, A024916, A072481, A237593, A244050, A245092, A262612, A274824, A319073, A340793, A345272, A353690.

Cf. A143128, A353908.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[sigma](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Oct 07 2021

Table[Sum[DivisorSigma[1, k] (n - k + 1), {k, n}], {n, 45}] (* Michael De Vlieger, Nov 24 2015 *)

a(n) = sum(x=1, n, sigma(x)*(n-x+1)) \\ Michel Marcus, Mar 18 2013

from math import isqrt
def A175254(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*(2*s+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+2*q+1)+3*(n+1)*(2*k+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

Conjecture: a(n) = Sum_{k=0..n} A006218(n-k). - R. J. Mathar, Oct 17 2012
a(n) = A000330(n) - A072481(n). - Omar E. Pol, Aug 12 2015
a(n) ~ Pi^2*n^3/36. - Vaclav Kotesovec, Sep 25 2016
G.f.: (1/(1 - x)^2)*Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) = Sum_{k=1..n} Sum_{i=1..k} k - (k mod i). - Wesley Ivan Hurt, Sep 13 2017
a(n) = A244050(n)/4. - Omar E. Pol, Jan 22 2021
a(n) = (n+1)*A024916(n) - A143128(n). - Vaclav Kotesovec, May 11 2022

Corrected by Jaroslav Krizek, Mar 17 2010

More terms from Michel Marcus, Mar 18 2013

A078181 a(n) = Sum_{d|n, d == 1 (mod 3)} d.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 8, 5, 1, 11, 1, 5, 14, 8, 1, 21, 1, 1, 20, 15, 8, 23, 1, 5, 26, 14, 1, 40, 1, 11, 32, 21, 1, 35, 8, 5, 38, 20, 14, 55, 1, 8, 44, 27, 1, 47, 1, 21, 57, 36, 1, 70, 1, 1, 56, 40, 20, 59, 1, 15, 62, 32, 8, 85, 14, 23, 68, 39, 1, 88, 1, 5, 74, 38, 26, 100, 8, 14, 80, 71, 1

Offset: 1

Vladeta Jovovic, Nov 21 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001817, A001822, A035382, A051731, A272716, A353908.

Cf. Sum_{d|n, d==1 mod k} d: A000593 (k=2), this sequence (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), A284099 (k=7), A284100 (k=8).

A078181 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) =1 then
            a :=a+d ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, May 11 2016

a[n_] := Plus @@ Select[Divisors[n], Mod[#, 3] == 1 &]; Array[a, 100] (* Giovanni Resta, May 11 2016 *)

G.f.: Sum_{n>=0} (3*n+1)*x^(3*n+1)/(1-x^(3*n+1)).
G.f.: -q*P'/P where P = Product_{n>=0} (1 - q^(3*n+1)). - Joerg Arndt, Aug 03 2011
Conjecture. If a(n)=n+1 then n==1 (mod 3). (Is this easy to settle? It has been verified for n=1,2,3,...,2000.) - John W. Layman, Apr 03 2006
The conjecture is false. The first and only counterexample below 10^8 is a(6800) = 6801 and 6800 == 2 (mod 3). - Lambert Herrgesell (zero815(AT)googlemail.com), May 06 2008
Equals A051731 * [1, 0, 0, 4, 0, 0, 7, 0, 0, 10, ...]. - Gary W. Adamson, Nov 06 2007
A272027(n/3) + a(n) + A078182(n) = A000203(n). - R. J. Mathar, May 25 2020
G.f.: Sum_{n >= 1} x^n*(1 + 2*x^(3*n))/(1 - x^(3*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Nov 26 2023

A078182 a(n) = Sum_{d|n, d == 2 (mod 3)} d.

Original entry on oeis.org

0, 2, 0, 2, 5, 2, 0, 10, 0, 7, 11, 2, 0, 16, 5, 10, 17, 2, 0, 27, 0, 13, 23, 10, 5, 28, 0, 16, 29, 7, 0, 42, 11, 19, 40, 2, 0, 40, 0, 35, 41, 16, 0, 57, 5, 25, 47, 10, 0, 57, 17, 28, 53, 2, 16, 80, 0, 31, 59, 27, 0, 64, 0, 42, 70, 13, 0, 87, 23, 56, 71, 10, 0, 76, 5, 40, 88, 28, 0, 115

Offset: 1

Vladeta Jovovic, Nov 21 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001817, A001822, A035386, A078181, A272715, A353908.

A078182 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) =2 then
            a :=a+d ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, May 11 2016

a[n_] := Plus @@ Select[Divisors[n], Mod[#, 3] == 2 &]; Array[a, 100] (* Giovanni Resta, May 11 2016 *)

a(n) = sumdiv(n, d, d*((d%3) == 2)); \\ Michel Marcus, May 11 2016

G.f.: Sum_{n>=0} (3*n+2)*x^(3*n+2)/(1-x^(3*n+2)).
A078181(n) + a(n) + 3*A000203(n/3) = A000203(n), where A000203 is defined as zero for non-integer arguments. - R. J. Mathar, May 11 2016
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Nov 26 2023

A186099 Sum of divisors of n congruent to 1 or 5 mod 6.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 31, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 57, 31, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38, 31, 20, 96, 14, 80, 6

Offset: 1

Michael Somos, Feb 12 2011

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f.: x + x^2 + x^3 + x^4 + 6*x^5 + x^6 + 8*x^7 + x^8 + x^9 + 6*x^10 + 12*x^11 +...
L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 +...
where exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 +...+ A003105(n)*x^n +...

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000593, A046913, A113957, A116073, A003105, A284098, A284104, A353908.

Cf. A004016, A005882, A005928.

Table[Total[Select[Divisors[n],MemberQ[{1,5},Mod[#,6]]&]],{n,0,100}]  (* Harvey P. Dale, Feb 24 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, If[ 1 == GCD[#, 6], #, 0] &]]; (* Michael Somos, Jun 27 2017 *)
a[ n_] := If[n < 1, 0, Times @@ (Which[# < 5, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)

{a(n) = sumdiv( n, d, d * (1 == gcd(d, 6) ))};

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n]};

a(n)=sigma(n/2^valuation(n,2)/3^valuation(n,3)) \\ Charles R Greathouse IV, Dec 07 2011

{S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
{a(n)=n*polcoeff(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n),n)}
for(n=1,80,print1(a(n),", "))
/* Paul D. Hanna, Feb 17 2013 */

Expansion of (1 + a(x)^2 - 2*a(x^2)^2) / 12 in powers of x where a() is a cubic AGM function.
a(n) is multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
Equals the logarithmic derivative of A003105, where A003105(n) = number of partitions of n into parts 6*n+1 or 6*n-1. - Paul D. Hanna, Feb 17 2013
L.g.f.: Sum_{n>=1} a(n)*x^n/n  =  Sum_{n>=1} S(n,x)*x^n/n  where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
a(n) = A284098(n) + A284104(n). - Seiichi Manyama, Mar 24 2017
G.f.: Sum_{n >= 1} x^n*(x^(10*n) + 5*x^(6*n) + 5*x^(4*n) + 1)/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021
From Amiram Eldar, Dec 30 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2/2^s)*(1-3/3^s).
Sum_{k=1..n} a(k) ~ c*n^2, where c = Pi^2/36 = 0.274155... (A353908). (End)

A280022 Expansion of phi_{5, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 48, 324, 1792, 3750, 15552, 19208, 61440, 85293, 180000, 175692, 580608, 399854, 921984, 1215000, 2031616, 1503378, 4094064, 2606420, 6720000, 6223392, 8433216, 6716184, 19906560, 12109375, 19192992, 21257640, 34420736, 21218430, 58320000, 29552672

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 23 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. this sequence (phi_{5, 4}), A280025 (phi_{7, 4}).
Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A282586 (E_2^3*E_4), A013974 (E_4*E_6 = E_10), A282431 (E_2^5).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), A282097 (n^2*sigma(n)), A282211 (n^3*sigma(n)), this sequence (n^4*sigma(n)).
Cf. A353908.

Table[n^4 * DivisorSigma[1, n], {n, 0, 32}] (* Amiram Eldar, Oct 31 2023 *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if(n < 1, 0, n^4 * sigma(n)); \\ Andrew Howroyd, Jul 23 2018

a(n) = n^4*A000203(n) for n > 0.
a(n) = (15*A282101(n) - 20*A282595(n) + 10*A282586(n) - 4*A013974(n) - A282431(n))/20736.
Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(4*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-5). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6. - Vaclav Kotesovec, Aug 02 2025

A086729 Decimal expansion of Pi^2/72.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

The original name was: Decimal expansion of Sum_{m=0..infinity} 1/(6*m+3)^2.

			0.1370778389040188697...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Index entries for transcendental numbers

Cf. A086722, A086723, A086724, A086725, A086726, A086727, A086728, A086730, A086731, A111003, A353908.

evalf(Pi^2/72) ;  # R. J. Mathar, Dec 18 2010

PARI
```
Pi^2/72 \\ Omar E. Pol, May 12 2022
```

Equals A111003/9. - R. J. Mathar, Dec 18 2010
From Amiram Eldar, Jul 19 2020: (Start)
Sum_{k>=0} (1/(12*k+3)^2 + 1/(12*k+9)^2).
Equals Integral_{x=1..oo} log(1 + 1/x^6)/x dx. (End)
Equals A353908/2. - Omar E. Pol, May 12 2022

New name after R. J. Mathar's Maple program. - Omar E. Pol, May 12 2022

A354238 Decimal expansion of 1 - Pi^2/12.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, May 20 2022

Ratio of area between the polygon that is adjacent in the same plane to the base of the stepped pyramid with an infinite number of levels described in A245092 and the circumscribed square (see the first formula).

			0.177532966575886781763792416676987405390525049396600781132220885314996264798...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013.

Ovidiu Furdui, Problem A, Nieuw Archief voor Wiskunde, Vol. 9, No. 1 (2008), p. 86; "Problem 2008/1-A, Solution to Problem A by Noud Aldenhoven and Daan Wanrooy, ibid., Vol. 9, No. 3 (2008), p. 303.
Ovidiu Furdui, Problem 1930, Mathematics Magazine, Vol. 86, No. 4 (2013), p. 289; A zeta series, Solution to Problem 1930 by Omran Kouba, ibid., Vol. 87, No. 4 (2014), pp. 296-298.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C7.4).
Index entries for transcendental numbers.

Cf. A000290, A004125, A013661, A024916, A072691, A152416, A237593, A245092, A353908.

RealDigits[1 - Pi^2/12, 10, 100][[1]] (* Amiram Eldar, May 20 2022 *)

PARI
```
1-Pi^2/12
```
PARI
```
1-zeta(2)/2
```

Equals lim_{n->infinity} A004125(n)/(n^2).
Equals 1 - A013661/2.
Equals 1 - A072691.
Equals A152416/2.
Equals Sum_{k>=1} 1/(2*k*(k+1)^2). - Amiram Eldar, May 20 2022
Equals -1/4 + Sum_{k>=2} (-1)^k * k * (k - Sum_{i=2..k} zeta(i)) (Furdui, 2013 problem). - Amiram Eldar, Jun 09 2022
Equals Integral_{x>=1} {x}/x^3 dx where {.} is the fractional part. [Nahin]. R. J. Mathar, May 22 2024
From Amiram Eldar, Jul 31 2025: (Start)
Equals Integral_{x=0..1} {1/x} * x dx (Furdui, 2013 book, section 2.21, page 103).
Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}*{y/x} dx dy, where {} denotes fractional part (Furdui, 2008 and 2013 book, section 2.36, page 105). (End)

A326125 Expansion of Sum_{k>=1} k^2 * x^k / (1 + x^k)^2.

Original entry on oeis.org

1, 2, 12, 4, 30, 24, 56, 8, 117, 60, 132, 48, 182, 112, 360, 16, 306, 234, 380, 120, 672, 264, 552, 96, 775, 364, 1080, 224, 870, 720, 992, 32, 1584, 612, 1680, 468, 1406, 760, 2184, 240, 1722, 1344, 1892, 528, 3510, 1104, 2256, 192, 2793, 1550, 3672, 728, 2862, 2160

Offset: 1

Ilya Gutkovskiy, Sep 10 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000593, A064987, A078306, A143520, A185152, A245579, A326238, A353908.

nmax = 54; CoefficientList[Series[Sum[k^2 x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(n/d + 1) d, {d, Divisors[n]}], {n, 1, 54}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n)={n*sumdiv(n, d, (-1)^(n/d+1)*d)} \\ Andrew Howroyd, Sep 10 2019

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^k * (1 + x^k) / (1 - x^k)^3.
a(n) = n * Sum_{d|n} (-1)^(n/d + 1) * d.
a(n) = n * A000593(n).
From Amiram Eldar, Dec 05 2022: (Start)
Multiplicative with a(2^e) = 2^e, and a(p^e) = p^e*(p^(e+1)-1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/36 = 0.2741556... (A353908). (End)
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(2-s)). - Amiram Eldar, Jan 07 2023

A333576 a(1) = 1; thereafter a(n) = n * uphi(n) / 2.

Original entry on oeis.org

1, 1, 3, 6, 10, 6, 21, 28, 36, 20, 55, 36, 78, 42, 60, 120, 136, 72, 171, 120, 126, 110, 253, 168, 300, 156, 351, 252, 406, 120, 465, 496, 330, 272, 420, 432, 666, 342, 468, 560, 820, 252, 903, 660, 720, 506, 1081, 720, 1176, 600, 816, 936, 1378, 702, 1100, 1176, 1026, 812, 1711, 720

Offset: 1

Ilya Gutkovskiy, Mar 27 2020

The unitary version of A023896.

Cf. A001221, A023896, A047994, A065464, A076479, A145388, A353908.

uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[1] = 1; a[n_] := n uphi[n]/2; Table[a[n], {n, 1, 60}]
a[n_] := (n/2) Sum[If[GCD[d, n/d] == 1, (-1)^PrimeNu[n/d] (d + 1), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

a(n) = if(n == 1, 1, my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - 1) * n / 2); \\ Amiram Eldar, Sep 21 2024

a(n) = (n/2) * Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * (d + 1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/36) * Product_{p prime} (1 - (2*p-1)/p^3) = A353908 * A065464 = 0.117407... . - Amiram Eldar, Sep 21 2024

cp's OEIS Frontend

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A086729 Decimal expansion of Pi^2/72.

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