A065608 -id:A065608 - cp's OEIS frontend

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

0, 1, 3, 7, 10, 19, 21, 35, 39, 56, 55, 91, 78, 113, 118, 155, 136, 208, 171, 252, 234, 287, 253, 395, 310, 404, 390, 497, 406, 614, 465, 651, 586, 698, 626, 910, 666, 875, 822, 1060, 820, 1202, 903, 1239, 1144, 1289, 1081, 1643, 1197, 1581, 1414, 1736

Offset: 1

Benoit Cloitre, Apr 08 2002

Inverse Mobius transform of A000217. - R. J. Mathar, Jan 19 2009

			x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A007437, A086666, A134840.

Cf. A032741, A065608, A363604, A363605, A363606.

with(numtheory):
seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # Peter Bala, Jan 21 2021

A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* Enrique Pérez Herrero, Feb 21 2012 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}

a(n) = my(f = factor(n)); (sigma(f, 2) - sigma(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - Vladeta Jovovic, Dec 17 2002
Row sums of triangle A134840. - Gary W. Adamson, Nov 12 2007
G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - Michael Somos, Feb 12 2008
G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - Michael Somos, Feb 12 2008
From Peter Bala, Jan 21 2021: (Start)
a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n)  A000203(n)) =  (1/2)*A086666.
G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 18 2001

			1.1373387363441965966969133683013475838308493098...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Harry J. Smith, Table of n, a(n) for n=1..2000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Eric Weisstein's World of Mathematics, Tree Searching.

Cf. A060867, A065442, A065608, A066766, A067616.

RealDigits[NSum[1/(2^k - 1)^2, {k, 1, Infinity}, PrecisionGoal -> 40, AccuracyGoal -> 40, WorkingPrecision -> 500, NSumTerms -> 50, NSumExtraTerms -> 50]][[1]] (* Peter Bertok (peter(AT)bertok.com), Dec 04 2001 *)
RealDigits[(Log[2] QPolyGamma[0, 1, 1/2] + QPolyGamma[1, 1, 1/2])/Log[2]^2 - 1, 10, 20][[1]] (* Eric W. Weisstein, Jun 02 2025 *)

{ default(realprecision, 2080); x=suminf(k=1, 1/(2^k - 1)^2); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065443.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

Equals Sum_{n>=1} 1/A060867(n).
From Amiram Eldar, Oct 16 2022: (Start)
Equals Sum_{k>=1} k/(2^(k+1)-1).
Equals A066766 - A065442. (End)
Equals Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 evaluated at q = 1/2 (see A065608). - Peter Bala, Oct 16 2022

More terms from Peter Bertok (peter(AT)bertok.com), Dec 04 2001

A137319 Start with the set of natural numbers. Add 1 to every 2nd term, 2 to every 3rd term, 3 to every 4th term, etc.

Original entry on oeis.org

1, 3, 5, 8, 9, 14, 13, 19, 19, 24, 21, 34, 25, 34, 35, 42, 33, 51, 37, 56, 49, 54, 45, 76, 53, 64, 63, 78, 57, 94, 61, 89, 77, 84, 79, 118, 73, 94, 91, 122, 81, 130, 85, 122, 117, 114, 93, 162, 103, 137, 119, 144, 105, 166, 123, 168, 133, 144, 117, 216, 121, 154, 161, 184

Offset: 1

Ctibor O. Zizka, Apr 06 2008

The generating function is the sum of the generating functions in A000027 and A065608. - R. J. Mathar, Apr 09 2008

			Start with the natural numbers:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
add 1 to every 2nd term:
   1,  3,  3,  5,  5,  7,  7,  9,  9, 11, ...
add 2 to every 3rd term:
   1,  3,  5,  5,  5,  9,  7,  9, 11, 11, ...
add 3 to every 4th term:
   1,  3,  5,  8,  5,  9,  7, 12, 11, 11, ...
add 4 to every 5th term:
   1,  3,  5,  8,  9,  9,  7, 12, 11, 15, ...
etc.

Cf. A000027, A065608.

A137319 := proc(n) local a,k ; a := n ; for k from 2 to n do if n mod k = 0 then a := a+k-1 ; fi ; od: a; end: seq(A137319(n),n=1..100) ; # R. J. Mathar, Apr 09 2008

Table[DivisorSigma[1, n] - DivisorSigma[0, n] + n, {n, 100}] (* Vincenzo Librandi, Sep 21 2015 *)

a(n) = sigma(n) - numdiv(n) + n; \\ Michel Marcus, Oct 29 2022

a(n) = n + A065608(n). - R. J. Mathar, Apr 09 2008

a(n) = Sum_{k=1..n} k^(1-ceiling(n/k)+floor(n/k)). - Wesley Ivan Hurt, May 24 2021

Corrected and extended by R. J. Mathar, Apr 09 2008

Edited by Jon E. Schoenfield, Sep 21 2015

A325940 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k)^2.

Original entry on oeis.org

0, 1, -2, 4, -4, 4, -6, 11, -10, 6, -10, 18, -12, 8, -20, 26, -16, 13, -18, 28, -28, 12, -22, 48, -28, 14, -36, 38, -28, 24, -30, 57, -44, 18, -44, 62, -36, 20, -52, 74, -40, 32, -42, 58, -72, 24, -46, 110, -54, 31, -68, 68, -52, 40, -68, 100, -76, 30, -58, 116

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A002129, A048272, A065608, A128315, A363615, A363616.

A325940:= func< n | (&+[0^(n mod j)*(-1)^j*(j-1): j in [1..n]]) >;
[A325940(n): n in [1..70]]; // G. C. Greubel, Jun 22 2024

nmax = 60; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^d (d - 1), {d, Divisors[n]}], {n, 1, 60}]

{a(n) = sumdiv(n, d, (-1)^d*(d-1))} \\ Seiichi Manyama, Sep 14 2019

def A325940(n): return sum(0^(n%j)*(-1)^j*(j-1) for j in range(1, n+1))
[A325940(n) for n in range(1,71)] # G. C. Greubel, Jun 22 2024

G.f.: Sum_{k>=2} (-1)^k * (k - 1) * x^k / (1 - x^k).
a(n) = Sum_{d|n} (-1)^d * (d - 1).
a(n) = A048272(n) - A002129(n).
Faster converging series: A(q) = Sum_{n >= 1} (-1)^n*q^(n^2)*((n-1)*q^(3*n) + n*q^(2*n) + (n-2)*q^n + n-1)/((1 + q^n)*(1 - q^(2*n))) - apply the operator t*d/dt to equation 1 in Arndt, then set t = -q and x = q. - Peter Bala, Jan 22 2021
a(n) = A128315(n, 2). - G. C. Greubel, Jun 22 2024

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Original entry on oeis.org

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A000292, A032741, A065608, A069153, A363605, A363606.
Cf. A059358, A363607, A363611.
Cf. A000203, A001158, A013662, A092348.

a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.
G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) - zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 4, 9, 19, 30, 52, 70, 107, 136, 191, 226, 314, 352, 463, 523, 664, 717, 919, 964, 1205, 1282, 1546, 1603, 1992, 2009, 2414, 2504, 2958, 2974, 3606, 3553, 4223, 4273, 4936, 4912, 5885, 5685, 6634, 6654, 7664, 7454, 8822, 8454, 9845

Offset: 0

N. J. A. Sloane, Feb 24 2001

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), A059821(k=4), A059822 (k=5), A059823 (k=6), A059824 (k=7), A059825 (k=8).
Cf. A002133, A065608, A191822, A191832.
Cf. A000203, A001157, A055507, A191829 (Andrews's D_{0,0,0}(n)), A191831 (Andrews's D_{0,1}(n)).

Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=3

D(x, y, n) = sum(k=1, n-1, sigma(k, x)*sigma(n-k, y));
D000(n) = sum(k=1, n-1, sigma(k, 0)*D(0, 0, n-k));
a(n) = if(n==0, 0, (3*D(0, 0, n)+3*D(0, 1, n)+D000(n)+2*sigma(n, 0)+3*sigma(n)+sigma(n, 2))/6); \\ Seiichi Manyama, Jul 26 2024

a(n) = ( 3*A055507(n-1) + 3*A191831(n) + A191829(n) + 2*sigma_0(n) + 3*sigma(n) + sigma_2(n) )/6. - Seiichi Manyama, Jul 26 2024

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686

Offset: 0

Gus Wiseman, Mar 28 2021

nonn

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.

			The a(2) = 1 through a(7) = 14 compositions:
  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
         (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
                (3,1)  (3,2)  (3,3)      (3,4)
                       (4,1)  (4,2)      (4,3)
                              (5,1)      (5,2)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]

\\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

Original entry on oeis.org

0, 1, 5, 16, 35, 76, 126, 226, 335, 531, 715, 1092, 1365, 1947, 2420, 3286, 3876, 5251, 5985, 7861, 8986, 11342, 12650, 16252, 17585, 21841, 24086, 29367, 31465, 38946, 40920, 49662, 53080, 62782, 66206, 80082, 82251, 97376, 102640, 120001, 123410, 146628

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013663, A032741, A065608, A069153, A363604, A363606.

Cf. A000203, A001157, A001158, A001159.

a[n_] := DivisorSum[n, Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^5)))

a(n) = my(f = factor(n)); (sigma(f, 4) + 2*sigma(f, 3) - sigma(f, 2) - 2*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+2,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+2,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) + 2*sigma_3(n) - sigma_2(n) - 2*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) + 2*zeta(s-3) - zeta(s-2) - 2*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

Original entry on oeis.org

0, 1, 6, 22, 56, 133, 252, 484, 798, 1344, 2002, 3157, 4368, 6441, 8630, 12112, 15504, 21274, 26334, 35014, 42762, 55133, 65780, 84349, 98336, 123124, 143304, 176373, 201376, 247380, 278256, 336744, 379000, 451402, 502250, 600055, 658008, 775733, 855042

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013664, A032741, A065608, A069153, A363604, A363605.

Cf. A000203, A001157, A001158, A001159, A001160.

a[n_] := DivisorSum[n, Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^6)))

a(n) = my(f = factor(n)); (sigma(f, 5) + 5*sigma(f, 4) + 5*sigma(f, 3) - 5*sigma(f, 2) - 6*sigma(f)) / 120; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+3,5) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+3,5).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 5*sigma_4(n) + 5*sigma_3(n) - 5*sigma_2(n) - 6*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 5*zeta(s-4) + 5*zeta(s-3) - 5*zeta(s-2) - 6*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 24, 29, 42, 45, 69, 66, 93, 98, 129, 120, 175, 153, 216, 206, 255, 231, 343, 282, 366, 354, 447, 378, 550, 435, 594, 542, 648, 582, 828, 630, 819, 770, 978, 780, 1114, 861, 1161, 1072, 1221, 1035, 1529, 1143, 1494, 1346, 1644, 1326, 1878, 1482, 1953

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A000005, A000203, A001157, A002117, A065608, A363611.

Cf. A007437, A069153.

a[n_] := DivisorSum[n, Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^3)))

a(n) = my(f = factor(n)); (sigma(f, 2) - 3*sigma(f) + 2*numdiv(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} binomial(k-1,2) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d-1,2).
From Amiram Eldar, Jan 01 2025: (Start)
a(n) = (sigma_2(n) - 3*sigma_1(n) + 2*sigma_0(n)) / 2.
Dirichlet g.f.: zeta(s) * (zeta(s-2) - 3*zeta(s-1) + 2*zeta(s)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

cp's OEIS Frontend

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

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A065443 Decimal expansion of Sum_{k=1..inf} 1/(2^k-1)^2.

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A137319 Start with the set of natural numbers. Add 1 to every 2nd term, 2 to every 3rd term, 3 to every 4th term, etc.

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A325940 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k)^2.

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A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

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A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

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A342532 Number of even-length compositions of n with alternating parts distinct.

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A059820 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) x^(t(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).