A113686 -id:A113686 - cp's OEIS frontend

A113685 Triangular array read by rows: T(n,k) is the number of partitions of n in which sum of odd parts is k, for k=0,1,...,n; n>=0.

1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 3, 0, 2, 0, 2, 0, 4, 0, 3, 0, 4, 0, 3, 0, 5, 5, 0, 3, 0, 4, 0, 4, 0, 6, 0, 5, 0, 6, 0, 6, 0, 5, 0, 8, 7, 0, 5, 0, 6, 0, 8, 0, 6, 0, 10, 0, 7, 0, 10, 0, 9, 0, 10, 0, 8, 0, 12, 11, 0, 7, 0, 10, 0, 12, 0, 12, 0, 10, 0, 15, 0, 11, 0, 14, 0, 15, 0

Offset: 0

Clark Kimberling, Nov 05 2005

(Sum over row n) = A000041(n) = number of partitions of n.
Reversal of this array is array in A113686.
From Gary W. Adamson, Apr 11 2010: (Start)
Let M = an infinite lower triangular matrix with A000041 interleaved with zeros: (1, 0, 1, 0, 2, 0, 3, 0, 5, ...) and Q = A000009 diagonalized with the rest zeros.
Then A113685 = M*Q. That row sums of the triangle (deleting prefaced zeros) = A000041 is equivalent to the identity: p(x) = p(x^2) * A000009(x). (End)

			First 5 rows:
  1;
  0, 1;
  1, 0, 1;
  0, 1, 0, 2;
  2, 0, 1, 0, 2;
  0, 2, 0, 2, 0, 3.
The partitions of 5 are 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1.
The sums of odd parts are 5,1,3,5,1,3,5, respectively, so that the numbers of 0's, 1's, 2s, 3s, 4s, 5s are 0,2,0,2,0,3, which is row 5 of the array.

Cf. A000041, A113686, A066967.

g := 1/product((1-t^(2*j-1)*x^(2*j-1))*(1-x^(2*j)),j=1..20):
gser := simplify(series(g,x=0,22)):
P[0] := 1: for n from 1 to 14 do P[n] := coeff(gser,x^n) od:
for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..n) od;
# yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

G.f.: G(t,x) = 1/Product_{j>=1} (1 - t^(2j-1)*x^(2j-1))*(1-x^(2j)). - Emeric Deutsch, Feb 17 2006

More terms from Emeric Deutsch, Feb 17 2006

A366528 Sum of odd prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2

Offset: 1

Gus Wiseman, Oct 22 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239(n).

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366531(n).

A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 70, 0, 0, 0, 165, 0, 0, 0, 330, 0, 0, 0, 704, 0, 0, 0, 1380, 0, 0, 0, 2688, 0, 0, 0, 4984, 0, 0, 0, 9394, 0, 0, 0, 16665, 0, 0, 0, 29970, 0, 0, 0, 52096, 0, 0, 0, 90090, 0, 0, 0, 152064, 0, 0, 0

Offset: 0

Clark Kimberling, Mar 13 2014

			a(8) counts these 4 partitions:  431, 41111, 3221, 221111.
From _Gus Wiseman_, Oct 24 2023: (Start)
The a(0) = 1 through a(12) = 12 partitions:
  ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)
                               (3221)             (651)
                               (41111)            (4332)
                               (221111)           (5421)
                                                  (33222)
                                                  (52221)
                                                  (63111)
                                                  (432111)
                                                  (3222111)
                                                  (6111111)
                                                  (42111111)
                                                  (222111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A239259, A239260, A239262, A239263, A000041.
The LHS (sum of odd parts) is counted by A113685.
The RHS (sum of even parts) is counted by A113686.
Without all the zeros we have a(4n) = A249914(n).
The strict case (without zeros) is A255001.
The Heinz numbers of these partitions are A366748, see also A019507.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.

z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
(* Peter J. C. Moses, Mar 12 2014 *)

A239260(n) + a(n) + A239262(n) = A000041(n).
From David A. Corneth, Oct 25 2023: (Start)
a(4*n) = A000009(2*n) * A000041(n) for n >= 0.
a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)

More terms from Alois P. Heinz, Mar 15 2014

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 3, 4, 3, 5, 5, 3, 4, 4, 6, 5, 6, 6, 5, 8, 7, 5, 6, 8, 6, 10, 7, 10, 9, 10, 8, 12, 11, 7, 10, 12, 12, 10, 15, 11, 14, 15, 15, 16, 12, 18, 15, 11, 14, 20, 18, 20, 15, 22, 15, 22, 21, 25, 24, 24, 18, 27

Offset: 0

Gus Wiseman, Oct 16 2023

The version for all k = 0..n is A113685 (including zeros).

			Triangle begins:
   1
   1
   1  1
   1  2
   2  1  2
   2  2  3
   3  2  2  4
   3  4  3  5
   5  3  4  4  6
   5  6  6  5  8
   7  5  6  8  6 10
   7 10  9 10  8 12
  11  7 10 12 12 10 15
  11 14 15 15 16 12 18
  15 11 14 20 18 20 15 22
  15 22 21 25 24 24 18 27
Row n = 8 counts the following partitions:
  (8)     (611)    (431)     (521)      (71)
  (62)    (4211)   (41111)   (332)      (53)
  (44)    (22211)  (3221)    (32111)    (5111)
  (422)            (221111)  (2111111)  (3311)
  (2222)                                (311111)
                                        (11111111)
Row n = 9 counts the following partitions:
  (81)     (63)      (54)       (72)        (9)
  (621)    (6111)    (522)      (5211)      (711)
  (441)    (432)     (4311)     (3321)      (531)
  (4221)   (42111)   (411111)   (321111)    (51111)
  (22221)  (3222)    (32211)    (21111111)  (333)
           (222111)  (2211111)              (33111)
                                            (3111111)
                                            (111111111)

Row sums are A000041.
The version including all k is A113685, even version A113686.
Column k = 1 is A119620.
The even version and the reverse version are both A174713.
For odd-indexed instead of odd parts we have A346697, even version A346698.
The corresponding rank statistic is A366528, even version A366531.
A000009 counts partitions into odd parts, ranks A066208.
A086543 counts partitions with odd parts, ranks A366322.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A045931, A053253, A066967, A130780, A171966, A241638, A268335, A325698, A366533.

Table[Length[Select[IntegerPartitions[n], Total[Select[#,OddQ]]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

T(n,k) = A000009(k) * A000041((n-k)/2).

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 4, 2, 2, 3, 5, 3, 4, 3, 6, 4, 4, 3, 5, 8, 5, 6, 6, 5, 10, 6, 8, 6, 5, 7, 12, 8, 10, 9, 10, 7, 15, 10, 12, 12, 10, 7, 11, 18, 12, 16, 15, 15, 14, 11, 22, 15, 20, 18, 20, 14, 11, 15

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.
The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
k-th triangle in the infinite set can be defined as having the sequence:
"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
All triangles in the infinite set have row sums = A000041.
The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
...
k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
...
k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
...
k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
...
Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

			First few rows of the triangle =
1;
1;
1, 1;
2, 1;
2, 1, 2;
3, 2, 2;
4, 2, 2, 3;
5, 3, 4, 3;
6, 4, 4, 3, 5;
8, 5, 6, 6, 5;
10, 6, 8, 6, 5, 7;
12, 8, 10, 9, 10, 7;
15, 10, 12, 12, 10, 7, 11;
18, 12, 16, 15, 15, 14, 11;
22, 15, 20, 18, 20, 14, 11, 15;
...
From _Gus Wiseman_, Oct 23 2023: (Start)
Row n = 9 counts the following partitions:
  (9)          (72)        (54)       (63)      (81)
  (711)        (5211)      (522)      (6111)    (621)
  (531)        (3321)      (4311)     (432)     (441)
  (51111)      (321111)    (411111)   (42111)   (4221)
  (333)        (21111111)  (32211)    (3222)    (22221)
  (33111)                  (2211111)  (222111)
  (3111111)
  (111111111)
(End)

Cf. A000009, A173305, A174712, A174714, A174715.
Row sums are A000041.
The odd version is A365067.
The corresponding rank statistic is A366531, odd version A366528.
A000009 counts partitions into odd parts, ranks A066208.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* Gus Wiseman, Oct 23 2023 *)

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

A366531 Sum of even prime indices of n.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 4, 0, 4, 0, 0, 2, 6, 4, 2, 0, 0, 4, 8, 0, 6, 0, 0, 2, 0, 6, 6, 4, 10, 2, 0, 0, 2, 0, 4, 4, 12, 8, 8, 0, 0, 6, 14, 0, 4, 0, 0, 2, 8, 0, 2, 6, 16, 6, 0, 4, 10, 10, 0, 2, 18, 0, 8, 0, 6, 2, 0, 0, 2, 4, 20, 4, 0, 12, 2, 8, 4, 8, 22, 0, 8, 0, 0, 6

Offset: 1

Gus Wiseman, Oct 22 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 198 are {1,2,2,5}, so a(198) = 2+2 = 4.

Zeros are A066208, counted by A000009.
The triangle for the odd version is A113685, without zeros A365067.
The triangle for this statistic is A113686, without zeros A174713.
The odd version is A366528.
The halved version is A366533.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A239261 counts partitions with sum of odd parts = sum of even parts.
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Table[Total[Cases[FactorInteger[n], {p_?(EvenQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366528(n).

A366533 Sum of even prime indices of n divided by 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 3, 2, 1, 0, 0, 2, 4, 0, 3, 0, 0, 1, 0, 3, 3, 2, 5, 1, 0, 0, 1, 0, 2, 2, 6, 4, 4, 0, 0, 3, 7, 0, 2, 0, 0, 1, 4, 0, 1, 3, 8, 3, 0, 2, 5, 5, 0, 1, 9, 0, 4, 0, 3, 1, 0, 0, 1, 2, 10, 2, 0, 6, 1, 4, 2, 4, 11, 0, 4, 0, 0, 3, 0, 7

Offset: 1

Gus Wiseman, Oct 23 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.

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

Zeros are A066208, counted by A000009.
The triangle for this statistic (without zeros) is A174713.
The un-halved odd version is A366528.
The un-halved version is A366531.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A365067 counts partitions by sum of odd parts (without zeros).
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

a(n) = A366531(n)/2.

A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

Original entry on oeis.org

72, 240, 672, 800, 2240, 4224, 5184, 6272, 9984, 14080, 17280, 33280, 39424, 48384, 52224, 57600, 93184, 116736, 161280, 174080, 192000, 247808, 304128, 373248, 389120, 451584, 487424, 537600, 565248, 585728, 640000, 718848, 1013760, 1089536, 1244160, 1384448

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

nonn

			6272 = 2*2*2*2*2*2*2*7*7 is droll since 2+2+2+2+2+2+2 = 14 = 7+7.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Giovanni Resta, Droll numbers, Numbers Aplenty.

For count instead of sum we have A072978.
Partitions of this type are counted by A239261, without zero terms A249914.
For prime indices instead of factors we have A366748, zeros of A366749.
The LHS is A366839 with alternating zeros, for indices A366531, triangle A113686.
The RHS is A366840, for indices A366528, triangle A113685.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A100367, A106529, A195017, A239241, A241638, A325698, A325700, A366533.

f:= proc(k, m) # numbers whose sum of prime factors >= m is k; m is prime
   local S,p,j;
   option remember;
   if k = 0 then return [1]
   elif m > k then return []
   fi;
   S:= NULL:
   p:= nextprime(m);
   for j from k by -m to 0 do
     S:= S, op(map(`*`,  procname(j,p) , m^((k-j)/m)))
   od;
   [S]
end proc:
g:= proc(N) local m,R;
  R:= NULL;
  for m from 1 while 2^m < N do
   R:= R, op(map(`*`,select(`<=`,f(2*m,3), N/2^m),2^m));
  od;
  sort([R])
end proc:
g(10^8); # Robert Israel, Feb 20 2025

Select[Range[2, 2*10^6, 2], First[#] == Total[Rest[#]] & [Times @@@ FactorInteger[#]] &] (* Paolo Xausa, Feb 19 2025 *)

isok(n) = {if (n % 2, return (0)); f = factor(n); return (2*f[1,2] == sum(i=2, #f~, f[i,1]*f[i,2]));} \\ Michel Marcus, Jun 21 2013

These are even numbers k such that A366839(k/2) = A366840(k). - Gus Wiseman, Oct 25 2023 (corrected Feb 19 2025)

Name edited by Paolo Xausa, Feb 19 2025

A066966 Total sum of even parts in all partitions of n.

Original entry on oeis.org

0, 2, 2, 10, 12, 30, 40, 82, 110, 190, 260, 422, 570, 860, 1160, 1690, 2252, 3170, 4190, 5760, 7540, 10142, 13164, 17450, 22442, 29300, 37410, 48282, 61170, 78132, 98310, 124444, 155582, 195310, 242722, 302570, 373882, 462954, 569130, 700570, 856970

Offset: 1

Vladeta Jovovic, Jan 26 2002

nonn

Partial sums of A206436. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 02 2023: (Start)
Convolution of A000041 and A146076.
Convolution of A002865 and A271342.
a(n) is also the sum of all even 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 even divisors are also all even parts of all partitions of n. (End)

			a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the even parts is 4+2+2+2 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Cf. A000041, A000203, A002865, A066186, A066897, A066898, A066967, A113686, A146076, A206436, A271342, A338156.

g:=sum(2*j*x^(2*j)/(1-x^(2*j)),j=1..55)/product(1-x^j,j=1..55): gser:=series(g,x=0,45): seq(coeff(gser,x^n),n=1..41);
# Emeric Deutsch, Feb 20 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ ((i+1) mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

max = 50; g = Sum[2*j*x^(2*j)/(1 - x^(2*j)), {j, 1, max}]/Product[1 - x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max - 1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], EvenQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

a(n) = 2*sum(k=1, floor(n/2), sigma(k)*numbpart(n-2*k) ); \\ Joerg Arndt, Jan 24 2014

a(n) = 2*Sum_{k=1..floor(n/2)} sigma(k)*numbpart(n-2*k).
a(n) = Sum_{k=0..n} k*A113686(n,k). - Emeric Deutsch, Feb 20 2006
G.f.: Sum_{j>=1} (2jx^(2j)/(1-x^(2j)))/Product_{j>=1}(1-x^j). - Emeric Deutsch, Feb 20 2006
a(n) = A066186(n) - A066967(n). - Omar E. Pol, Mar 10 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

More terms from Emeric Deutsch, Feb 20 2006

A366749 Self-signed alternating sum of the prime indices of n.

Original entry on oeis.org

0, -1, 2, -2, -3, 1, 4, -3, 4, -4, -5, 0, 6, 3, -1, -4, -7, 3, 8, -5, 6, -6, -9, -1, -6, 5, 6, 2, 10, -2, -11, -5, -3, -8, 1, 2, 12, 7, 8, -6, -13, 5, 14, -7, 1, -10, -15, -2, 8, -7, -5, 4, 16, 5, -8, 1, 10, 9, -17, -3, 18, -12, 8, -6, 3, -4, -19, -9, -7, 0

Offset: 1

Gus Wiseman, Oct 23 2023

sign

We define the self-signed alternating sum of a multiset y to be Sum_{k in y} k*(-1)^k.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

With summands of 2^(n-1) we get A048675.
With summands of (-1)^k we get A195017.
The version for alternating prime indices is A346697 - A346698 = A316524.
Positions of zeros are A366748, counted by A239261.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A300061 lists numbers with even sum of prime indices, odd A300063.
A366528 adds up odd prime indices, counted by A113685.
A366531 adds up even prime indices, counted by A113686.
Cf. A000720, A019507, A045931, A055396, A061395, A325698, A325700, A366533.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
asum[y_]:=Sum[x*(-1)^x,{x,y}];
Table[asum[prix[n]],{n,100}]

a(n) = Sum_{k in A112798(n)} k*(-1)^k.

a(n) = A366531(n) - A366528(n).

cp's OEIS Frontend

A113685 Triangular array read by rows: T(n,k) is the number of partitions of n in which sum of odd parts is k, for k=0,1,...,n; n>=0.

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A366528 Sum of odd prime indices of n.

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A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

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A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

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A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

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A366531 Sum of even prime indices of n.

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A366533 Sum of even prime indices of n divided by 2.

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A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

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