A341062 -id:A341062 - cp's OEIS frontend

A006218 a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n.

0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, 113, 119, 123, 127, 131, 140, 142, 146, 150, 158, 160, 168, 170, 176, 182, 186, 188, 198, 201, 207, 211, 217, 219, 227, 231, 239, 243, 247, 249

Offset: 0

N. J. A. Sloane

The identity Sum_{k=1..n} floor(n/k) = Sum_{k=1..n} d(k) is Equation (10), p. 58, of Apostol (1976). - N. J. A. Sloane, Dec 06 2020
The "Dirichlet divisor problem" is to find a precise asymptotic estimate for this sequence - see formula lines below, also Apostol (1976), Chap. 3.
Number of increasing arithmetic progressions where n+1 is the second or later term. - Mambetov Timur, Takenov Nurdin, Haritonova Oksana (timus(AT)post.kg; oksanka-61(AT)mail.ru), Jun 13 2002. E.g., a(3) = 5 because there are 5 such arithmetic progressions: (1, 2, 3, 4); (2, 3, 4); (1, 4); (2, 4); (3, 4).
Binomial transform of A001659.
Area covered by overlapped partitions of n, i.e., sum of maximum values of the k-th part of a partition of n into k parts. - Jon Perry, Sep 08 2005
Equals inverse Mobius transform of A116477. - Gary W. Adamson, Aug 07 2008
The Polymath project (see the Tao-Croot-Helfgott link) sketches an algorithm for computing a(n) in essentially cube root time, see section 2.1. - Charles R Greathouse IV, Oct 10 2010 [Sladkey gives another. - Charles R Greathouse IV, Oct 02 2017]
The Dirichlet inverse starts (offset 1) 1, -3, -5, 1, -10, 16, -16, 1, 2, 33, -29, -6, -37, 55, 55, -1, -52, -5, -60, ... - R. J. Mathar, Oct 17 2012
The inverse Mobius transforms yields A143356. - R. J. Mathar, Oct 17 2012
An improved approximation vs. Dirichlet is: a(n) = log(Gamma(n+1)) + 2n*gamma. Using sample ranges of {n = k^2-k to k^2 + (k-1)} the means of the new error term are < +- 0.5 up to k=150, except on two values of k. These ranges appear to give means closest to zero for such small sample sizes. It is not clear sample means remain < +- 0.5 at larger k. The standard deviations are ~(n*log(n))^(1/4)/2, with n near sample range center. - Richard R. Forberg, Jan 06 2015
The values of n for which a(n) is even are given by 4*m^2 <= n <= 4*m(m+1) for m >= 0. Example: for m=1 the values of n are 4 <= n <= 8 for which a(4) to a(8) are even. - G. C. Greubel, Sep 30 2015
For n > 0, a(n) = count(x|y), 1 <= y <= x <= n, that is, the number of pairs in the ordered list of x and y, where y divides x, up to and including n. - Torlach Rush, Jan 31 2017
a(n) is also the total number of partitions of all positive integers <= n into equal parts. - Omar E. Pol, May 29 2017
a(n) is the rank of the join of the set of elements of rank n in Young's lattice, the lattice of all integer partitions ordered by inclusion of their Ferrers diagrams. - Geoffrey Critzer, Jul 11 2018
a(n) always has the same parity as floor(sqrt(n)) = A000196(n): see A211264 (proof in Diophante link). - Bernard Schott, Feb 13 2021
From Omar E. Pol, Feb 16 2021: (Start)
Apart from initial zero this is the convolution of A341062 and A000027.
Nonzero terms convolved with A341062 gives A055507. (End)
From Bernard Schott, Apr 17 2022: (Start)
a(n-1) is the number of lattice points in the first quadrant lying under the hyperbola x*y = n, excluding the lattice points on the axes.
a(n) is the number of lattice points in the first quadrant lying on or under the hyperbola x*y = n, excluding the lattice points on the axes. (Reference Hari Kishan). (End)
Let tiles Tn (for n >= 1) be initially placed on square n on an infinite 1D board. At each step, the leftmost unblocked tile (i.e., the top tile in the leftmost stack) jumps forward exactly n squares. Tiles can stack, and only the top tile of a stack can move. This sequence gives the step number when tile n moves for the first time. - Ali Sada, May 23 2025

			a(3) = 5 because 3 + floor(3/2) + 1 = 3 + 1 + 1 = 5. Or tau(1) + tau(2) + tau(3) = 1 + 2 + 2 = 5.
a(4) = 8 because 4 + floor(4/2) + floor(4/3) + 1 = 4 + 2 + 1 + 1 = 8. Or
tau(1) + tau(2) + tau(3) + tau(4) = 1 + 2 + 2 + 3 = 8.
a(5) = 10 because 5 + floor(5/2) + floor(5/3) + floor (5/4) + 1 = 5 + 2 + 1 + 1 + 1 = 10. Or tau(1) + tau(2) + tau(3) + tau(4) + tau(5) = 1 + 2 + 2 + 3 + 2 = 10.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
K. Chandrasekharan, Introduction to Analytic Number Theory. Springer-Verlag, 1968, Chap. VI.
K. Chandrasekharan, Arithmetical Functions. Springer-Verlag, 1970, Chapter VIII, pp. 194-228. Springer-Verlag, Berlin.
P. G. L. Dirichlet, Werke, Vol. ii, pp. 49-66.
M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972, p. 7.
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
Hari Kishan, Number Theory, Krishna, Educational Publishers, 2014, Theorem 1, p. 133.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 56.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Nurdin N. Takenov and Oksana Haritonova, Representation of positive integers by a special set of digits and sequences, in Dolmatov, S. L. et al. editors, Materials of Science, Practical seminar "Modern Mathematics".
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.6.13 on page 107.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 1000 terms from T. D. Noe)
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanța, Vol. 22(1),2014, 13-23.
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340. See Eq. 1.5.
D. Berkane, O. Bordellès, and O. Ramaré, Explicit upper bounds for the remainder term in the divisor problem, Math. Comp. 81:278 (2012), pp. 1025-1051.
Peter J. Cameron and Hamid Reza Dorbidi, Minimal cover groups, arXiv:2311.15652 [math.GR], 2023. See p. 13.
Diophante, A1712, La même parité (in French).
Xiaoxi Duan and M. W. Wong, The Dirichlet divisor problem, traces and determinants for complex powers of the twisted bi-Laplacian, J. of Math. Analysis and Applications, Volume 410, Issue 1, Feb 01 2014, Pages 151-157
L. Hoehn and J. Ridenhour, Summations involving computer-related functions, Math. Mag., 62 (1989), 191-196.
M. N. Huxley, Exponential Sums and Lattice Points III, Proc. London Math. Soc., 87 (2003), pp. 591-609.
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Richard Sladkey, A successive approximation algorithm for computing the divisor summatory function, arXiv:1206.3369 [math.NT], 2012.
Terence Tao, Ernest Croot III, and Harald Helfgott, Deterministic methods to find primes, Math. Comp. 81 (2012), 1233-1246; also at arXiv:1009.3956 [math.NT], 2010-2012.

Right edge of A056535. Cf. A000005, A001659, A052511, A143236.
Row sums of triangle A003988, A010766 and A143724.
A061017 is an inverse.
It appears that the partial sums give A078567. - N. J. A. Sloane, Nov 24 2008
Cf. A116477, A051731, A055507, A161700, A004125, A212120, A341062.

a006218 n = sum $ map (div n) [1..n]
-- Reinhard Zumkeller, Jan 29 2011

[0] cat [&+[Floor(n/k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019

with(numtheory): A006218 := n->add(sigma[0](i), i=1..n);

Table[Sum[DivisorSigma[0, k], {k, n}], {n, 70}]
FoldList[Plus, 0, Table[DivisorSigma[0, x], {x, 61}]] //Rest (* much faster *)
Join[{0},Accumulate[DivisorSigma[0,Range[60]]]] (* Harvey P. Dale, Jan 06 2016 *)

PARI
```
a(n)=sum(k=1,n,n\k)
```

a(n)=sum(k=1,sqrtint(n),n\k)*2-sqrtint(n)^2 \\ Charles R Greathouse IV, Oct 10 2010

from sympy import integer_nthroot
def A006218(n): return 2*sum(n//k for k in range(1,integer_nthroot(n,2)[0]+1))-integer_nthroot(n,2)[0]**2 # Chai Wah Wu, Mar 29 2021

a(n) = n * ( log(n) + 2*gamma - 1 ) + O(sqrt(n)), where gamma is the Euler-Mascheroni number ~ 0.57721... (see A001620), Dirichlet, 1849. Again, a(n) = n * ( log(n) + 2*gamma - 1 ) + O(log(n)*n^(1/3)). The determination of the precise size of the error term is an unsolved problem (the so-called Dirichlet divisor problem) - see references, especially Huxley (2003).
The bounds from Chandrasekharan lead to the explicit bounds n log(n) + (2 gamma - 1) n - 4 sqrt(n) - 1 <= a(n) <= n log(n) + (2 gamma - 1) n + 4 sqrt(n). - David Applegate, Oct 14 2008
a(n) = 2*(Sum_{i=1..floor(sqrt(n))} floor(n/i)) - floor(sqrt(n))^2. - Benoit Cloitre, May 12 2002
G.f.: (1/(1-x))*Sum_{k >= 1} x^k/(1-x^k). - Benoit Cloitre, Apr 23 2003
For n > 0: A027750(a(n-1) + k) = k-divisor of n, = k <= A000005(n). - Reinhard Zumkeller, May 10 2006
a(n) = A161886(n) - n + 1 = A161886(n-1) - A049820(n) + 2 = A161886(n-1) + A000005(n) - n + 2 = A006590(n) + A000005(n) - n = A006590(n+1) - n - 1 = A006590(n) + A000005(n) - n for n >= 2. a(n) = a(n-1) + A000005(n) for n >= 1. - Jaroslav Krizek, Nov 14 2009
D(n) = Sum_{m >= 2, r >= 1} (r/m^(r+1)) * Sum_{j = 1..m - 1} * Sum_{k = 0 .. m^(r+1) - 1} exp{ 2*k*pi i(p^n + (m - j)m^r) / m^(r+1) } where p is some fixed prime number. - A. Neves, Oct 04 2010
Let E(n) = a(n) - n(log n + 2 gamma - 1). Then Berkane-Bordellès-Ramaré show that |E(n)| <= 0.961 sqrt(n), |E(n)| <= 0.397 sqrt(n) for n > 5559, and |E(n)| <= 0.764 n^(1/3) log n for x > 9994. - Charles R Greathouse IV, Jul 02 2012
a(n) = Sum_{k = 1..floor(sqrt(n))} A005408(floor((n/k) - (k-1))). - Gregory R. Bryant, Apr 20 2013
Dirichlet g.f. for s > 2: Sum_{n>=1} a(n)/n^s = Sum_{k>=1} (Zeta(s-1) - Sum_{n=1..k-1} (HurwitzZeta(s,n/k)*n/k^s))/k. - Mats Granvik, Sep 24 2017
From Ridouane Oudra, Dec 31 2022: (Start)
a(n) = n^2 - Sum_{i=1..n} Sum_{j=1..n} floor(log(i*j)/log(n+1));
a(n) = floor(sqrt(n)) + 2*Sum_{i=1..n} floor((sqrt(i^2 + 4*n) - i)/2);
a(n) = n + Sum_{i=1..n} v_2(i)*round(n/i), where v_2(i) = A007814(i). (End)

A006128 Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 12, 20, 35, 54, 86, 128, 192, 275, 399, 556, 780, 1068, 1463, 1965, 2644, 3498, 4630, 6052, 7899, 10206, 13174, 16851, 21522, 27294, 34545, 43453, 54563, 68135, 84927, 105366, 130462, 160876, 198014, 242812, 297201, 362587, 441546, 536104, 649791, 785437, 947812, 1140945, 1371173, 1644136, 1968379, 2351597, 2805218, 3339869, 3970648, 4712040, 5584141, 6606438, 7805507, 9207637

Offset: 0

N. J. A. Sloane, Colin Mallows

a(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989).) - Wolfdieter Lang
Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g., for n=3 to n=2 we have a(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, see A066186. - Thomas Wieder, May 20 2004
Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel, Nov 07 2005
Starting with offset 1, = the partition triangle A008284 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008
Starting with offset 1 equals A000041: (1, 1, 2, 3, 5, 7, 11, ...) convolved with A000005: (1, 2, 2, 3, 2, 4, ...). - Gary W. Adamson, Jun 16 2009
Apart from initial 0 row sums of triangle A066633, also the Möbius transform is A085410. - Gary W. Adamson, Mar 21 2011
More generally, the total number of parts >= k in all partitions of n equals the sum of k-th largest parts of all partitions of n. In this case k = 1. Apart from initial 0 the first column of A181187. - Omar E. Pol, Feb 14 2012
Row sums of triangle A221530. - Omar E. Pol, Jan 21 2013
From Omar E. Pol, Feb 04 2021: (Start)
a(n) is also the total number of divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.
Apart from initial zero this is also as follows:
Convolution of A000005 and A000041.
Convolution of A006218 and A002865.
Convolution of A341062 and A000070.
Row sums of triangles A221531, A245095, A339258, A340525, A340529. (End)
Number of ways to choose a part index of an integer partition of n, i.e., partitions of n with a selected position. Selecting a part value instead of index gives A000070. - Gus Wiseman, Apr 19 2021

			For n = 4 the partitions of 4 are [4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]. The total number of parts is 12. On the other hand, the sum of the largest parts of all partitions is 4 + 2 + 3 + 2 + 1 = 12, equaling the total number of parts, so a(4) = 12. - _Omar E. Pol_, Oct 12 2018

S. M. Luthra, On the average number of summands in partitions of n, Proc. Nat. Inst. Sci. India Part. A, 23 (1957), p. 483-498.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Paul Erdős and Joseph Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8, (1941), 335-345.
John A. Ewell, Additive evaluation of the divisor function, Fibonacci Quart. 45 (2007), no. 1, 22-25. See Table 1.
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008; see p.27
I. Kessler and M. Livingston, The expected number of parts in a partition of n, Monatsh. Math. 81 (1976), no. 3, 203-212.
I. Kessler and M. Livingston, The expected number of parts in a partition of n, Monatsh. Math. 81 (1976), no. 3, 203-212.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, Graph - The asymptotic ratio
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
S. M. Luthra, On the average number of summands in partitions of n, Proc. Nat. Inst. Sci. India Part. A, 23 (1957), p. 483-498.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
C. L. Mallows & N. J. A. Sloane, Emails, Jun. 1991
Ljuben Mutafchiev, On the Largest Part Size and Its Multiplicity of a Random Integer Partition, arXiv:1712.03233 [math.PR], 2017.
Omar E. Pol, Illustration of initial terms
J. Sandor, D. S. Mitrinovic, B. Crstici, Handbook of Number Theory I, Volume 1, Springer, 2005, p. 495.
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.
H. S. Wilf, A unified setting for selection algorithms (II), Annals Discrete Math., 2 (1978), 135-148.

Cf. A093694, A008284, A000005, A066633, A085410, A046746, A209423, A285220, A336902, A336903.
Main diagonal of A210485.
Column k=1 of A256193.
The version for normal multisets is A001787.
The unordered version is A001792.
The strict case is A015723.
The version for factorizations is A066637.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A336875 counts compositions with a selected part.
A339564 counts factorizations with a selected factor.
Cf. A066186, A083710, A130689, A264401, A276024, A343341.

List([0..60],n->Length(Flat(Partitions(n)))); # Muniru A Asiru, Oct 12 2018

a006128 = length . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

g:= add(n*x^n*mul(1/(1-x^k), k=1..n), n=1..61):
a:= n-> coeff(series(g,x,62),x,n):
seq(a(n), n=0..61);
# second Maple program:
a:= n-> add(combinat[numbpart](n-j)*numtheory[tau](j), j=1..n):
seq(a(n), n=0..61);  # Alois P. Heinz, Aug 23 2019

a[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; Table[ a[n], {n, 0, 41}]
CoefficientList[ Series[ Sum[n*x^n*Product[1/(1 - x^k), {k, n}], {n, 100}], {x, 0, 100}], x]
a[n_] := Plus @@ Max /@ IntegerPartitions@ n; Array[a, 45] (* Robert G. Wilson v, Apr 12 2011 *)
Join[{0}, ((Log[1 - x] + QPolyGamma[1, x])/(Log[x] QPochhammer[x]) + O[x]^60)[[3]]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Length /@ Table[IntegerPartitions[n] // Flatten, {n, 50}] (* Shouvik Datta, Sep 12 2021 *)

f(n)= {local(v,i,k,s,t);v=vector(n,k,0);v[n]=2;t=0;while(v[1]1,i--;s+=i*(v[i]=(n-s)\i));t+=sum(k=1,n,v[k]));t } /* Thomas Baruchel, Nov 07 2005 */

a(n) = sum(m=1, n, numdiv(m)*numbpart(n-m)) \\ Michel Marcus, Jul 13 2013

from sympy import divisor_count, npartitions
def a(n): return sum([divisor_count(m)*npartitions(n - m) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 25 2017

G.f.: Sum_{n>=1} n*x^n / Product_{k=1..n} (1-x^k).
G.f.: Sum_{k>=1} x^k/(1-x^k) / Product_{m>=1} (1-x^m).
a(n) = Sum_{k=1..n} k*A008284(n, k).
a(n) = Sum_{m=1..n} of the number of divisors of m * number of partitions of n-m.
Note that the formula for the above comment is a(n) = Sum_{m=1..n} d(m)*p(n-m) = Sum_{m=1..n} A000005(m)*A000041(n-m), if n >= 1. - Omar E. Pol, Jan 21 2013
Erdős and Lehner show that if u(n) denotes the average largest part in a partition of n, then u(n) ~ constant*sqrt(n)*log n.
a(n) = A066897(n) + A066898(n), n>0. - Reinhard Zumkeller, Mar 09 2012
a(n) = A066186(n) - A196087(n), n >= 1. - Omar E. Pol, Apr 22 2012
a(n) = A194452(n) + A024786(n+1). - Omar E. Pol, May 19 2012
a(n) = A000203(n) + A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{m=1..p(n)} A194446(m) = Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1. - Omar E. Pol, May 12 2013
a(n) = A198381(n) + A026905(n), n >= 1. - Omar E. Pol, Aug 10 2013
a(n) = O(sqrt(n)*log(n)*p(n)), where p(n) is the partition function A000041(n). - Peter Bala, Dec 23 2013
a(n) = Sum_{m=1..n} A006218(m)*A002865(n-m), n >= 1. - Omar E. Pol, Jul 14 2014
From Vaclav Kotesovec, Jun 23 2015: (Start)
Asymptotics (Luthra, 1957): a(n) = p(n) * (C*N^(1/2) + C^2/2) * (log(C*N^(1/2)) + gamma) + (1+C^2)/4 + O(N^(-1/2)*log(N)), where N = n - 1/24, C = sqrt(6)/Pi, gamma is the Euler-Mascheroni constant A001620 and p(n) is the partition function A000041(n).
The formula a(n) = p(n) * (sqrt(3*n/(2*Pi)) * (log(n) + 2*gamma - log(Pi/6)) + O(log(n)^3)) in the abstract of the article by Kessler and Livingston (cited also in the book by Sandor, p. 495) is incorrect!
Right is: a(n) = p(n) * (sqrt(3*n/2)/Pi * (log(n) + 2*gamma - log(Pi^2/6)) + O(log(n)^3))
or a(n) ~ exp(Pi*sqrt(2*n/3)) * (log(6*n/Pi^2) + 2*gamma) / (4*Pi*sqrt(2*n)).
(End)
G.f.: (log(1-x) + psi_x(1))/(log(x) * (x)inf), where psi_q(z) is the q-digamma function, and (q)_inf is the q-Pochhammer symbol (the Euler function). - _Vladimir Reshetnikov, Nov 17 2016
a(n) = Sum_{m=1..n} A341062(m)*A000070(n-m), n >= 1. - Omar E. Pol, Feb 05 2021 2014

A138137 First differences of A006128.

Original entry on oeis.org

1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133

Offset: 1

Omar E. Pol, Mar 18 2008

Number of parts in the last section of the set of partitions of n (see A135010, A138121).
Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011
From Omar E. Pol, Feb 16 2021: (Start)
Convolution of A341062 and A000041.
Convolution of A000005 and A002865.
a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)

			From _Omar E. Pol_, Feb 19 2012: (Start)
Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
.                                           6
.                                           3+3
.                                           4+2
.                                           2+2+2
.                              5              1
.                              3+2              1
.                    4           1              1
.                    2+2           1              1
.            3         1           1              1
.      2       1         1           1              1
.  1     1       1         1           1              1
. --- ----- ------- --------- ----------- --------------
.  1,  2,    3,      6,        8,          15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
From _Omar E. Pol_, Aug 16 2013: (Start)
The geometric model looks like this:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    2      3        6          8           15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - _Omar E. Pol_, Jul 27 2021

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 1 of A207031.

Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)

a(n) = A006128(n) - A006128(n-1).
a(n) = A000041(n-1) + A138135(n). - Omar E. Pol, Nov 07 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016
G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017

A055507 a(n) = Sum_{k=1..n} d(k)*d(n+1-k), where d(k) is number of positive divisors of k.

Original entry on oeis.org

1, 4, 8, 14, 20, 28, 37, 44, 58, 64, 80, 86, 108, 108, 136, 134, 169, 160, 198, 192, 236, 216, 276, 246, 310, 288, 348, 310, 400, 344, 433, 396, 474, 408, 544, 450, 564, 512, 614, 522, 688, 560, 716, 638, 756, 636, 860, 676, 859, 772, 926, 758, 1016, 804, 1032

Offset: 1

Leroy Quet, Jun 29 2000

nonn

a(n) is the number of ordered ways to express n+1 as a*b+c*d with 1 <= a,b,c,d <= n. - David W. Wilson, Jun 16 2003
tau(n) (A000005) convolved with itself, treating this result as a sequence whose offset is 2. - Graeme McRae, Jun 06 2006
Convolution of A341062 and nonzero terms of A006218. - Omar E. Pol, Feb 16 2021

			a(4) = d(1)*d(4) + d(2)*d(3) + d(3)*d(2) + d(4)*d(1) = 1*3 +2*2 +2*2 +3*1 = 14.
3 = 1*1+2*1 in 4 ways, so a(2)=4; 4 = 1*1+1*3 (4 ways) = 2*1+2*1 (4 ways), so a(3)=8; 5 = 4*1+1*1 (4 ways) = 2*2+1*1 (2 ways) + 3*1+2*1 (8 ways), so a(4) = 14. - _N. J. A. Sloane_, Jul 07 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See D_{0,0}.
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.
Yoichi Motohashi, The binary additive divisor problem, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 27 no. 5 (1994), p. 529-572.
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.

Cf. A000005, A000385, A072031, A212151.

Cf. A006218, A341062.

with(numtheory); A055507:=n->add(tau(j)*tau(n+1-j),j=1..n);

Table[Sum[DivisorSigma[0, k]*DivisorSigma[0, n + 1 - k], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 08 2022 *)

a(n)=sum(k=1,n,numdiv(k)*numdiv(n+1-k)) \\ Charles R Greathouse IV, Oct 17 2012

from sympy import divisor_count
def A055507(n): return  (sum(divisor_count(i+1)*divisor_count(n-i) for i in range(n>>1))<<1)+(divisor_count(n+1>>1)**2 if n&1 else 0) # Chai Wah Wu, Jul 26 2024

G.f.: Sum_{i >= 1, j >= 1} x^(i+j-1)/(1-x^i)/(1-x^j). - Vladeta Jovovic, Nov 11 2001
Working with an offset of 2, it appears that the o.g.f is equal to the Lambert series sum {n >= 2} A072031(n-1)*x^n/(1 - x^n). - Peter Bala, Dec 09 2014
a(n) = A212151(n+2) - A212151(n+1). - Ridouane Oudra, Sep 12 2020

More terms from James Sellers, Jul 04 2000

Definition clarified by N. J. A. Sloane, Jul 07 2012

A078567 Number of arithmetic subsequences of [1..n] with length > 1.

Original entry on oeis.org

0, 1, 4, 9, 17, 27, 41, 57, 77, 100, 127, 156, 191, 228, 269, 314, 364, 416, 474, 534, 600, 670, 744, 820, 904, 991, 1082, 1177, 1278, 1381, 1492, 1605, 1724, 1847, 1974, 2105, 2245, 2387, 2533, 2683, 2841, 3001, 3169, 3339, 3515, 3697, 3883, 4071, 4269, 4470

Offset: 1

Robert E. Sawyer (rs.1(AT)mindspring.com)

The number of arithmetic subsequences of [1..n] with successive-term increment i and length k is n-i*(k-1) for i > 0, k > 0, n > i*(k-1).
Appears to be the partial sums of A006218. - N. J. A. Sloane, Nov 24 2008
The O(n^(1/2)) formula can be derived via Dirichlet hyperbola method (see Wikipedia link below) applied to a(n) = Sum_{k=1..n-1} Sum_{i*j=k} (sqrt(n)*sqrt(n)-i*j), where we've written the formula in this form to show which functions are being Dirichlet convoluted. - Daniel Hoying, May 31 2020
Apart from initial zero this is the convolution of A341062 and the nonzero terms of A000217. - Omar E. Pol, Feb 16 2021

			a(2): [1,2]; a(3): [1,2],[1,3],[2,3],[1,2,3].

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Wikipedia, Dirichlet Hyperbola Method

Cf. A000005, A000027, A000217, A006218, A051336, A134546, A341062.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[tau](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Oct 07 2021

a[n_]:=-(-1 + n) n + Sum[-(1/2) Ceiling[n/(1 + k)] (-1 - k - 2 n + (1 + k) Ceiling[n/(1 + k)]), {k, 0, n - 2}]; (* Lorenz H. Menke, Jr., Feb 17 2017 *)
Table[Sum[(n - i) DivisorSigma[0, i], {i, n}], {n, 47}] (* or *)
With[{nn = 46}, {0}~Join~Table[First[ListConvolve @@ Transpose@ Take[#, n]], {n, nn}] &@ Table[{n, DivisorSigma[0, n]}, {n, nn}]] (* Michael De Vlieger, Feb 18 2017 *)

a(n)=sum(i=1,n, numdiv(i)*(n-i)) \\ Charles R Greathouse IV, Feb 18 2017

a(n)={n--; sqrtint(n)^2*(1/4 * (1+sqrtint(n))^2-n-1) + sum(i=1, sqrtint(n), (n\i)*(2*n + 2 - i*(1+n\i)))} \\ Andrew Howroyd, May 31 2020

from math import isqrt
def A078567(n):
    m = isqrt(n-1)
    return m**2*(1+m)**2//4-m**2*n+sum((n-1)//i*(2*n-i*(1+(n-1)//i)) for i in range(1,m+1)) # Chai Wah Wu, Oct 07 2021

a(n) = Sum_{i=1..n-1} Sum_{j=1..floor((n-1)/i)} (n - i*j).
Convolution of A000027 and A000005. - Vladeta Jovovic, Apr 08 2006
Row sums of triangle A134546. - Gary W. Adamson, Oct 31 2007
a(n) = Sum_{i=1..n} (n-i) * A000005(i). - Wesley Ivan Hurt, May 08 2016
G.f.: (x/(1 - x)^2)*Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{k=1..n-1} Sum_{i=1..n-1} floor(k/i). - Wesley Ivan Hurt, Sep 14 2017
a(n) = Sum_{k=1..n-1} Sum_{i|k} (n-k). - Daniel Hoying, May 26 2020
a(n+1) = floor(sqrt(n))^2*(1/4*(1+floor(sqrt(n)))^2 - n - 1) + Sum_{i=1..floor(sqrt(n))} floor(n/i)*(2*n + 2 - i*(1+floor(n/i))). - Daniel Hoying, May 31 2020

A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 39, 54, 71, 94, 124, 159, 201, 258, 322, 401, 499, 613, 750, 918, 1110, 1340, 1617, 1935, 2308, 2752, 3261, 3854, 4554, 5350, 6273, 7348, 8572, 9983, 11612, 13460, 15578, 18007, 20761, 23894, 27473, 31511, 36090, 41296, 47152, 53767

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A000005 and A000009.

Apart from initial zero this is the convolution of A341062 and A036469. - Omar E. Pol, Feb 16 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A000009, A006128, A015723, A209423.

Cf. A001227, A048272, A067588, A090867, A036469, A341062.

nmax = 50; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[((Log[1-x] + QPolyGamma[0, 1, x]) * QPochhammer[-1, x]) / (2*Log[x]), {x, 0, nmax}], x]

a(n) ~ 3^(1/4)*(2*gamma + log(12*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620.

A191831 a(n) = Sum_{i+j=n, i,j >= 1} tau(i)*sigma(j), where tau() = A000005(), sigma() = A000203().

Original entry on oeis.org

0, 1, 5, 12, 24, 39, 60, 87, 113, 158, 189, 249, 286, 372, 402, 516, 545, 696, 709, 886, 912, 1125, 1110, 1401, 1348, 1674, 1654, 1992, 1906, 2390, 2226, 2735, 2648, 3141, 2926, 3705, 3346, 4069, 3898, 4604, 4223, 5282, 4707, 5757, 5426, 6326, 5754, 7269, 6324, 7669, 7230, 8468, 7556, 9456, 8240, 10018, 9320, 10748, 9621, 12246

Offset: 1

N. J. A. Sloane, Jun 17 2011

This is Andrews's D_{0,1}(n).
From Omar E. Pol, Dec 08 2021: (Start)
Zero together with the convolution of A000005 and A000203.
Zero together with the convolution of A341062 and A024916.
Zero together with the convolution of the nonzero terms of A006218 and A340793.
a(n) is also the volume of a symmetric polycube which belongs to the family of symmetric polycubes that represent the convolution of A000203 with any other integer sequence, n >= 1. (End)

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130.

Cf. A000005, A000203 A006218, A024916, A086718, A237593, A340793, A341062.

with(numtheory); D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
[seq(D01(n),n=1..60)];

Table[Sum[DivisorSigma[0, j] DivisorSigma[1, n - j], {j, n - 1}], {n, 60}] (* Michael De Vlieger, Jan 01 2017 *)

a(n)=sum(i=1,n-1,numdiv(i)*sigma(n-i)) \\ Charles R Greathouse IV, Feb 19 2013

G.f.: (Sum_{k>=1} x^k/(1 - x^k))*(Sum_{k>=1} k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Jan 01 2017

A284870 Expansion of Sum_{i>=1} ix^i/(1 - x) Product_{j=1..i} 1/(1 - x^j).

Original entry on oeis.org

0, 1, 4, 10, 22, 42, 77, 131, 217, 345, 537, 812, 1211, 1767, 2547, 3615, 5078, 7043, 9687, 13185, 17815, 23867, 31766, 41972, 55146, 71997, 93519, 120813, 155358, 198811, 253374, 321509, 406436, 511802, 642264, 803140, 1001154, 1243966, 1541167, 1903754, 2345300, 2881404, 3531195, 4316632, 5264444, 6405389

Offset: 0

Ilya Gutkovskiy, Apr 04 2017

nonn

Total number of parts in all partitions of all positive integers <= n.
Sum of largest parts of all partitions of all positive integers <= n.
From Omar E. Pol, Feb 16 2021: (Start)
Apart from initial zero this is as follows:
Convolution of A341062 and A014153.
Convolution of A000005 and A000070.
Convolution of nonzero terms of A006218 and A000041.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A340581. These divisors are also all parts of all partitions of all positive integers <= n. (End)

			a(4) = 22 because we have 1 = 1, 2 = 2, 1 + 1 = 2, 3 = 3, 2 + 1 = 3, 1 + 1 + 1 = 3, 4 = 4, 3 + 1 = 4, 2 + 2 = 4, 2 + 1 + 1 = 4 and 1 + 1 + 1 + 1 = 4 therefore 1 + 1 + 2 + 1 + 2 + 3 + 1 + 2 + 2 + 3 + 4 = 22 (total number of parts) or 1 + 2 + 1 + 3 + 2 + 1 + 4 + 3 + 2 + 2 + 1 = 22 (sum of largest parts).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000070, A006128, A015724, A026905, A124920, A138137, A182738, A299779.

Cf. A000041, A006218, A014153, A340581, A341062.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1) +(p-> p+[0, p[1]])(b(n-i, min(n-i, i))))
    end:
a:= proc(n) a(n):= `if`(n<1, 0, a(n-1)+b(n$2)[2]) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Feb 16 2021

nmax = 45; CoefficientList[Series[Sum[i x^i /(1 - x) Product[1/(1 - x^j), {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[1/(1 - x) Sum[x^i /(1 - x^i), {i, 1, nmax}] Product[1/(1 - x^j), {j, 1, nmax}], {x, 0, nmax}], x]
Accumulate[Table[Sum[DivisorSigma[0, k] PartitionsP[n - k], {k, 1, n}], {n, 0, 45}]]

G.f.: Sum_{i>=1} i*x^i/(1 - x) * Product_{j=1..i} 1/(1 - x^j).
G.f.: (1/(1 - x)) * Sum_{i>=1} x^i/(1 - x^i) * Product_{j>=1} 1/(1 - x^j).
a(n) = Sum_{k=0..n} A006128(k).
a(n) = A124920(n+1) - 1.
a(n) = Sum_{k=1..n} k * A299779(n,k). - Alois P. Heinz, May 14 2018

cp's OEIS Frontend

A006218 a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n.

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A006128 Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n.

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A138137 First differences of A006128.

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A055507 a(n) = Sum_{k=1..n} d(k)*d(n+1-k), where d(k) is number of positive divisors of k.

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A078567 Number of arithmetic subsequences of [1..n] with length > 1.

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A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

A284870 Expansion of Sum_{i>=1} ix^i/(1 - x) Product_{j=1..i} 1/(1 - x^j).