A002541 -id:A002541 - cp's OEIS frontend

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

1, 1, 2, 4, 5, 9, 10, 16, 19, 26, 27, 44, 45, 57, 65, 87, 88, 120, 121, 158, 171, 200, 201, 278, 284, 331, 353, 426, 427, 536, 537, 646, 676, 766, 782, 982, 983, 1106, 1154, 1365, 1366, 1617, 1618, 1851, 1943, 2146, 2147, 2589, 2600, 2917, 3008, 3390, 3391

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

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

Cf. A002541, A003238, A010766, A014668, A126656, A167865, A316782, A317712, A320222, A320224.

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,30}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A320225(n): return 1 if n == 1 else sum(A320225(d)*(n//d-1) for d in range(1,n)) # Chai Wah Wu, Jun 08 2022

a(1) = 1; a(n > 1) = Sum_{d = 1..n-1} a(d) * floor(n/d-1).

G.f. A(x) satisfies A(x) = x + (1/(1 - x)) * Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

A320226 Number of integer partitions of n whose non-1 parts are all equal.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 15, 18, 19, 24, 25, 28, 31, 35, 36, 41, 42, 47, 50, 53, 54, 61, 63, 66, 69, 74, 75, 82, 83, 88, 91, 94, 97, 105, 106, 109, 112, 119, 120, 127, 128, 133, 138, 141, 142, 151, 153, 158, 161, 166, 167, 174, 177, 184, 187, 190, 191, 202

Offset: 0

Gus Wiseman, Oct 07 2018

nonn

			The integer partitions whose non-1 parts are all equal:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (61)       (44)
             (111)  (31)    (221)    (51)      (331)      (71)
                    (211)   (311)    (222)     (511)      (611)
                    (1111)  (2111)   (411)     (2221)     (2222)
                            (11111)  (2211)    (4111)     (3311)
                                     (3111)    (22111)    (5111)
                                     (21111)   (31111)    (22211)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.

Cf. A002541, A003238, A070776, A126656, A316782, A317712, A320224, A320225.

Table[Length[Select[IntegerPartitions[n],SameQ@@DeleteCases[#,1]&]],{n,30}]

a(n > 1) = A002541(n - 1) + 1.

A024917 a(n) = Sum_{k=2..n} k*floor(n/k).

Original entry on oeis.org

2, 5, 11, 16, 27, 34, 48, 60, 77, 88, 115, 128, 151, 174, 204, 221, 259, 278, 319, 350, 385, 408, 467, 497, 538, 577, 632, 661, 732, 763, 825, 872, 925, 972, 1062, 1099, 1158, 1213, 1302, 1343, 1438, 1481, 1564, 1641, 1712, 1759, 1882, 1938, 2030, 2101, 2198, 2251

Offset: 2

Clark Kimberling

nonn

Cf. A002541, A024916, A345682.

[&+[k*Floor(n/k): k in [2..n]]: n in [2..55]]; // Bruno Berselli, Jan 08 2012

Table[Sum[k*Floor[n/k],{k,2,n}],{n,2,60}] (* Harvey P. Dale, Mar 13 2015 *)

a(n) = sum(k=2,n, k*floor(n/k)); \\ Michel Marcus, Sep 02 2019

from math import isqrt
def A024917(n): return (-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n # Chai Wah Wu, Oct 23 2023

G.f.: (1/(1 - x)) * Sum_{k>=2} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 02 2019

A134867 A010766 * A000012.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 8, 4, 2, 1, 10, 5, 3, 2, 1, 14, 8, 5, 3, 2, 1, 16, 9, 6, 4, 3, 2, 1, 20, 12, 8, 6, 4, 3, 2, 1, 23, 14, 10, 7, 5, 4, 3, 2, 1, 27, 17, 12, 9, 7, 5, 4, 3, 2, 1, 29, 18, 13, 10, 8, 6, 5, 4, 3, 2, 1, 35, 23, 17, 13, 10, 8, 6, 5, 4, 3, 2, 1, 37, 24, 18, 14, 11, 9, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

			First few rows of the triangle:
   1;
   3,  1;
   5,  2,  1;
   8,  4,  2, 1;
  10,  5,  3, 2, 1;
  14,  8,  5, 3, 2, 1;
  16,  9,  6, 4, 3, 2, 1;
  20, 12,  8, 6, 4, 3, 2, 1;
  23, 14, 10, 7, 5, 4, 3, 2, 1;
  27, 17, 12, 9, 7, 5, 4, 3, 2, 1;
  ...

Column k=1..4 give: A006218, A002541, A366968, A366972.
Row sums give A024916.
Cf. A010766, A135539.

t = Table[Sum[Floor[n/h], {h, k, n}], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A134867 array *)
TableForm[t]    (* A134867 sequence *)
(* Clark Kimberling, Oct 11 2014 *)

T(n, k) = sum(j=k, n, n\j); \\ Seiichi Manyama, Oct 30 2023

A010766 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial row sums of A010766 starting fromt the right.
G.f. of column k: 1/(1-x) * Sum_{j>=1} x^(k*j)/(1-x^j) = 1/(1-x) * Sum_{j>=k} x^j/(1-x^j). - Seiichi Manyama, Oct 30 2023

More terms from Seiichi Manyama, Oct 30 2023

A264402 Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 4, 1, 4, 5, 2, 2, 8, 4, 1, 4, 9, 7, 2, 3, 12, 10, 4, 1, 4, 14, 15, 7, 2, 2, 17, 20, 12, 4, 1, 6, 18, 27, 17, 7, 2, 2, 23, 33, 26, 12, 4, 1, 4, 24, 44, 35, 19, 7, 2, 4, 27, 51, 49, 28, 12, 4, 1, 5, 30, 64, 63, 41, 19, 7, 2, 2

Offset: 1

Emeric Deutsch, Nov 21 2015

T(n,k) = number of partitions of n in which the 2nd largest part is k (0 if all parts are equal). Example: T(7,2) = 4 because we have [3,2,1,1], [3,2,2], [4,2,1], and [5,2].
The fact that the above two statistics (in Name and in 1st Comment) have the same distribution follows at once by conjugation. - Emeric Deutsch, Dec 11 2015
Row sums yield the partition numbers (A000041).
T(n,0) = A000005(n) = number of divisors of n.
Sum_{k>=0} k*T(n,k) = A182984(n).

			T(7,2) = 4 because we have [2,2,1,1,1], [3,2,1,1], [3,3,1], and [4,2,1].
Triangle starts:
1;
2;
2,1;
3,2;
2,4,1;
4,5,2;
2,8,4,1;

Alois P. Heinz, Rows n = 1..350, flattened

Cf. A000005, A000041, A116685, A182984, A002541.

g := sum(x^i/((1-x^i)*(product(1-t*x^j, j = i+1 .. 100))), i = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n to 27 do P[n] := sort(coeff(gser, x, n)) end do: for n to 27 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, b(n, i-1) +add((p->[0, p[1]+
       expand(p[2]*x^j)])(b(n-i*j, i-1)) , j=1..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)[2]):
seq(T(n), n=1..20);  # Alois P. Heinz, Nov 29 2015

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Sum[ Function[p, {0, p[[1]] + Expand[p[[2]]*x^j]}][b[n-i*j, i-1]], {j, 1, n/i} ]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n][[2]]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

G.f.: G(t,x) = Sum_{i>=1} (x^i/((1 - x^i)*Product_{j>=i+1}(1-t*x^j))).

A275495 a(n) = Sum_{k=2..n} floor(n/k) - 2floor(n/(2k)).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 4, 6, 7, 8, 7, 8, 9, 12, 10, 11, 12, 13, 12, 15, 16, 17, 14, 16, 17, 20, 19, 20, 21, 22, 19, 22, 23, 26, 24, 25, 26, 29, 26, 27, 28, 29, 28, 33, 34, 35, 30, 32, 33, 36, 35, 36, 37, 40, 37, 40, 41, 42, 39, 40, 41, 46, 42, 45, 46, 47, 46, 49

Offset: 1

Peter Luschny, Jul 30 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002541, row sums of A275510, A059851.

seq(add(floor(n/k)-2*floor(n/(2*k)), k=2..n), n=1..60); # Ridouane Oudra, Oct 20 2019

Table[Sum[Floor[n/k] - 2*Floor[n/(2*k)], {k, 2, n}], {n, 1, 50}] (* G. C. Greubel, Jul 30 2016 *)

a(n)=sum(k=2,n,n\k) - 2*sum(k=2,n\2,n\(2*k)) \\ Charles R Greathouse IV, Jul 30 2016

[sum([floor(n/k) - 2*floor(n/(2*k)) for k in (2..n)]) for n in (1..69)]

a(n) = Sum_{i=1..n} floor((n-i)/i)*(-1)^(i+1). - Wesley Ivan Hurt, Sep 13 2017
a(n) = Sum_{i=2..n} (floor(n/i) mod 2) = A059851(n) - (n mod 2). - Ridouane Oudra, Oct 20 2019
a(n) ~ log(2) * n. - Vaclav Kotesovec, May 28 2021

A284755 Numbers n such that the average of all proper divisors of all positive integers <= n is an integer.

Original entry on oeis.org

2, 3, 63, 1249, 4696, 1200509

Offset: 1

Ilya Gutkovskiy, Apr 02 2017

Numbers n such that A002541(n)|A153485(n).

a(7) > 10^12. - Giovanni Resta, Apr 13 2017

Eric Weisstein's World of Mathematics, Restricted Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A001065, A002541, A003601, A023884, A023886, A027751, A153485, A226647, A284288.

Select[Range[2, 1300000], Mod[Sum[DivisorSigma[1, k] - k, {k, 1, #}], Sum[DivisorSigma[0, k] - 1, {k, 1, #}]] == 0 &]

A339370 a(n) = Sum_{k=1..floor(n/2)} (n-k) * floor((n-k)/k).

Original entry on oeis.org

0, 1, 4, 11, 19, 36, 50, 76, 102, 138, 165, 227, 262, 318, 381, 460, 510, 614, 672, 791, 889, 990, 1064, 1249, 1353, 1477, 1610, 1790, 1891, 2133, 2244, 2455, 2626, 2798, 2983, 3312, 3452, 3649, 3857, 4198, 4356, 4715, 4883, 5190, 5514, 5763, 5949, 6446, 6686, 7045

Offset: 1

Wesley Ivan Hurt, Dec 01 2020

Total area of all y X z rectangles such that x + y = n, with x and y integers, 0 < x <= y and z = floor(y/x). - Wesley Ivan Hurt, Dec 21 2020

Cf. A002541, A153485.

Table[Sum[(n - k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 60}]

a(n) = sum(k=1, n\2, (n-k) * ((n-k)\k)); \\ Michel Marcus, Dec 02 2020

from math import isqrt
def A339370(n): return n*(1-n)+(s:=isqrt(n))**2*(s+1-(n<<1))-sum((q:=n//k)*((k-(n<<1)<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 27 2023

a(n) ~ n^2*(log(n) + 2*EulerGamma - Pi^2/12 - 3/2). - Rok Cestnik, Dec 20 2020

a(n) = n*A002541(n) - A153485(n). - Vaclav Kotesovec, Dec 21 2020

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 37, 37, 38, 39, 39, 40, 42, 42, 43, 44, 46, 46

Offset: 1

Gus Wiseman, Jul 23 2022

nonn

For initial terms up to 30 we have a(n) = Log_2 A355537(n).

The sum of the same range is A000096.
The product of the same range is A000142, Heinz number A070826.
For divisors (not just prime factors) we get A002541, also A006218, A077597.
A shifted variation is A013939.
The unshifted version is A022559, product A327486, w/o multiplicity A355537.
The ranges themselves are the rows of A131818 (shifted).
Partial sums of A297155 (shifted).
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A066843 gives partial sums of A000005.
Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

a(n) = A013939(n) - n + 1.

A156745 a(n) = Sum_{k=1..n} floor((n+k)/k) = n + Sum_{k=1..n} sigma_0(k), where sigma_0(k) is A000005(k). Also a(n) = n + A006218(n).

Original entry on oeis.org

2, 5, 8, 12, 15, 20, 23, 28, 32, 37, 40, 47, 50, 55, 60, 66, 69, 76, 79, 86, 91, 96, 99, 108, 112, 117, 122, 129, 132, 141, 144, 151, 156, 161, 166, 176, 179, 184, 189, 198, 201, 210, 213, 220, 227, 232, 235, 246, 250, 257, 262, 269, 272, 281, 286, 295, 300

Offset: 1

Ctibor O. Zizka, Feb 14 2009

Generalized sequence b(n) = Sum_{k=1..n} floor((n+k*t)/k) = t*n + Sum_{k=1..n} sigma_0(k), where sigma_0(k) is A000005(k). Also b(n) = t*n + A006218(n).

Partial sums of A334954. - Omar E. Pol, Sep 26 2020

Cf. A000005, A006218, A153818, A118014, A002541, A334954.

a(n) = n + sum(k=1, n, numdiv(k)); \\ Michel Marcus, Oct 02 2020

from math import isqrt
def A156745(n): return n-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023

a(n) = 2*n + Sum_{k=1..floor(n/2)} floor((n-k)/k). - Wesley Ivan Hurt, Dec 25 2020

a(n) = A005843(n) + A002541(n), after Wesley Ivan Hurt. - Omar E. Pol, Dec 25 2020

More terms from Eric M. Schmidt, Feb 28 2014

cp's OEIS Frontend

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

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A320226 Number of integer partitions of n whose non-1 parts are all equal.

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A024917 a(n) = Sum_{k=2..n} k*floor(n/k).

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Magma

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A134867 A010766 * A000012.

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A264402 Triangle read by rows: T(n,k) is the number of partitions of n that have k parts larger than the smallest part (n>=1, k>=0).

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Maple

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A275495 a(n) = Sum_{k=2..n} floor(n/k) - 2*floor(n/(2*k)).

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A284755 Numbers n such that the average of all proper divisors of all positive integers <= n is an integer.

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Mathematica

A339370 a(n) = Sum_{k=1..floor(n/2)} (n-k) * floor((n-k)/k).

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A275495 a(n) = Sum_{k=2..n} floor(n/k) - 2floor(n/(2k)).