A015723 -id:A015723 - cp's OEIS frontend

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

1, 15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

After 1, first differs from A318992 in lacking 390, with prime indices {1,2,3,6}.
First differs from A343337 in having 195, with prime indices {2,3,6}.
Alternative name: 1 and numbers where either the smallest prime index is not a divisor of all the other prime indices, or the greatest prime index is not divisible by all the other prime indices.
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.
Also Heinz numbers of partitions that either empty, have smallest part not dividing all the others, or have greatest part not divisible by all the others (counted by A343346). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            90: {1,2,2,3}      141: {2,15}
     15: {2,3}         91: {4,6}          143: {5,6}
     30: {1,2,3}       93: {2,11}         145: {3,10}
     33: {2,5}         95: {3,8}          150: {1,2,3,3}
     35: {3,4}         99: {2,2,5}        153: {2,2,7}
     45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
     51: {2,7}        105: {2,3,4}        155: {3,11}
     55: {3,5}        110: {1,3,5}        161: {4,9}
     60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
     66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
     69: {2,9}        123: {2,13}         175: {3,3,4}
     70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
     75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
     77: {4,5}        138: {1,2,9}        182: {1,4,6}
     85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
For example, the prime indices of 90 are {1,2,2,3}, and, because 1 divides all the other parts, 90 is in the sequence, even though 3 is not divisible by all the other parts.

The partitions without these Heinz numbers are counted by A130714.
The first condition alone gives A342193.
The second condition alone gives A343337.
The "and" instead of "or" version is A343338.
The partitions with these Heinz numbers are counted by A343346.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A083710, A257993, A338470, A339562, A341450, A343339, A343340, A343341, A343342, A343378, A343379, A343382.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)||!And@@IntegerQ/@(p/Min@@p)]&]

Equals the union of A342193 and A343337.

A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 3, 4, 7, 8, 9, 10, 15, 16, 18, 23, 30, 32, 35, 42, 51, 59, 63, 73, 89, 100, 106, 125, 145, 160, 174, 198, 229, 255, 274, 310, 355, 388, 420, 472, 534, 582, 631, 701, 784, 859, 928, 1021, 1144, 1243, 1338, 1475, 1630, 1767, 1909, 2089, 2299

Offset: 0

Naohiro Nomoto, Feb 01 2002

T. D. Noe, Table of n, a(n) for n = 0..1000
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A000700, A000701, A006128, A015723, A046682, A079499, A158691.

CoefficientList[Series[Sum[n*q^(2n-1)*Product[1+q^k, {k, 1, 2n-3, 2}], {n, 1, 30}], {q, 0, 60}], q]

G.f.: A(q) = Sum_{n >= 1} n*q^(2*n-1)*(1+q)*(1+q^3)*...*(1+q^(2*n-3)).
From Peter Bala, Aug 20 2017: (Start)
Let F(q) = Product_{i >= 1} (1 + q^(2*i-1)). Then A(q) = Sum_{n >= 0} ( F(q) - Product_{i = 1..n} (1 + q^(2*i-1)) ).
It follows that the above sum A(q) satisfies -A(q-1) = 1 + q + 3*q^2 + 12*q^3 + 61*q^4 + ..., the g.f. for A158691, row-Fishburn matrices of size n. (End)

Edited by Dean Hickerson, Feb 11 2002

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.

A325505 Heinz number of the set of Heinz numbers of all strict integer partitions of n.

Original entry on oeis.org

2, 3, 5, 143, 493, 62651, 26718511, 22017033127, 44220524211551, 52289759420183033963, 546407750301194131199484983, 8362548333129019658779663581495109, 1828111016191440393570169991636207115709029581, 1059934964500839879758659437301868941873808925011368355891

Offset: 0

Gus Wiseman, May 07 2019

nonn

The Heinz number of a set or sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Also Heinz numbers of rows of A246867 (squarefree numbers arranged by sum of prime indices A056239).

			The strict integer partitions of 5 are {(5), (4,1), (3,2)}, with Heinz numbers {11,14,15}, with Heinz number prime(11)*prime(14)*prime(15) = 62651, so a(6) = 62651.
The sequence of terms together with their prime indices begins:
                            2: {1}
                            3: {2}
                            5: {3}
                          143: {5,6}
                          493: {7,10}
                        62651: {11,14,15}
                     26718511: {13,21,22,30}
                  22017033127: {17,26,33,35,42}
               44220524211551: {19,34,39,55,66,70}
         52289759420183033963: {23,38,51,65,77,78,105,110}
  546407750301194131199484983: {29,46,57,85,91,102,130,154,165,210}

Index entries for sequences related to Heinz numbers

Cf. A001222, A003963, A015723, A056239, A066189, A112798, A145519, A147655, A215366, A246867, A325500 (non-strict version), A325504, A325506, A325512, A325513.

Table[Times@@Prime/@(Times@@Prime/@#&/@Select[IntegerPartitions[n],UnsameQ@@#&]),{n,7}]

a(n) = Product_{i = 1..A000009(n)} prime(A246867(n,i)).
A001221(a(n)) = A001222(a(n)) = A000009(n).
A056239(a(n)) = A147655(n).
A003963(a(n)) = A325506(n).

A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 3, 1, 0, 1, 1, 7, 8, 5, 1, 0, 1, 1, 9, 16, 15, 8, 1, 0, 1, 1, 11, 27, 35, 28, 10, 1, 0, 1, 1, 13, 41, 69, 73, 49, 13, 1, 0, 1, 1, 15, 58, 121, 160, 170, 86, 18, 1, 0, 1, 1, 17, 78, 195, 311, 460, 357, 156, 25, 1

Offset: 0

Alois P. Heinz, Sep 19 2019

Row n is binomial transform of the n-th row of triangle A327632.

			Square array A(n,k) begins:
  0,  0,  0,   0,    0,    0,    0,     0,     0, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  1,  1,   1,    1,    1,    1,     1,     1, ...
  1,  3,  5,   7,    9,   11,   13,    15,    17, ...
  1,  3,  8,  16,   27,   41,   58,    78,   101, ...
  1,  5, 15,  35,   69,  121,  195,   295,   425, ...
  1,  8, 28,  73,  160,  311,  553,   918,  1443, ...
  1, 10, 49, 170,  460, 1047, 2106,  3865,  6611, ...
  1, 13, 86, 357, 1119, 2893, 6507, 13182, 24625, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A057427, A015723, A327605, A327628.
Rows n=0,(1+2),3-5 give: A000004, A000012, A005408, A104249, A005894.
Main diagonal gives: A327623.
Cf. A327618, A327632.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
    end:
A:= (n, k)-> b(n$2, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n==0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i(i + 1)/2 < n, Return@{0, 0}]; b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2020, after Maple *)

A(n,k) = Sum_{i=0..k} binomial(k,i) * A327632(n,i).

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006

Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012

			T(8,3)=2 because we have [5,3] and [4,3,1].
Triangle begins:
n/k 1 2 3 4 5 6 7 8 9 10
01: 1
02: 0 1
03: 1 1 1
04: 1 0 1 1
05: 1 1 1 1 1
06: 2 2 1 1 1 1
07: 2 2 1 2 1 1 1
08: 3 2 2 1 2 1 1 1
09: 3 3 3 2 2 2 1 1 1
10: 5 4 4 3 2 2 2 1 1 1
...
The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), which is row n = 6. - _Gus Wiseman_, May 07 2019

Mircea Merca, Table of n, a(n) for n = 1..7260

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.

Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

g:=product(1+x^j,j=1..50)*sum(t^i*x^i/(1+x^i),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 29 2006
seq(seq(coeff(x^k*(product(1+x^j, j=1..n))/(1+x^k), x, n), k=1..n), n=1..13); # Mircea Merca, Feb 28 2014

z = 15; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; s[n_] := s[n] = Flatten[Table[p[n, k], {k, 1, PartitionsQ[n]}]]; t[n_, k_] := Count[s[n], k]; u = Table[t[n, k], {n, 1, z}, {k, 1, n}]; TableForm[u] (* A015716 as a triangle *)
v = Flatten[u] (* A015716 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)

G.f.: G(t,x) = Product_{j>=1} (1+x^j) * Sum_{i>=1} t^i*x^i/(1+x^i). - Emeric Deutsch, Mar 29 2006
From Mircea Merca, Feb 28 2014: (Start)
a(n) = A238450(n) + A238451(n).
T(n,k) = Sum_{j=1..floor(n/k)} (-1)^(j-1)*A000009(n-j*k).
G.f.: for column k: q^k/(1+q^k)*(-q;q)_{inf}. (End)

A079499 Total number of parts in all partitions of n into distinct odd parts.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 1, 4, 4, 4, 4, 6, 7, 6, 10, 12, 13, 12, 16, 18, 22, 22, 25, 32, 36, 36, 42, 50, 53, 58, 64, 76, 83, 88, 99, 116, 123, 132, 147, 168, 181, 194, 215, 240, 262, 280, 306, 346, 375, 396, 437, 482, 521, 558, 610, 670, 724, 772, 840, 922, 993, 1056, 1151, 1256, 1348

Offset: 0

Arnold Knopfmacher, Jan 21 2003

nonn

Also sum of the sizes of the Durfee squares of all self-conjugate partitions of n. Example: a(13)=7 because there are three self-conjugate partitions of 13, namely [7,1,1,1,1,1,1], [5,3,3,1,1] and [4,4,3,2], having Durfee squares of sizes 1,3 and 3, respectively. a(n) = Sum_{k=1..floor(sqrt(n))} k*A116422(n,k). - Emeric Deutsch, Feb 14 2006

			a(13)=7 because the partitions of 13 into distinct odd parts are [13], [9,3,1] and [7,5,1] and we have 1+3+3=7 parts.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..1000 from T. D. Noe)
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A015723, A000700, A067619, A006128.
Cf. A032021.
Cf. A116422.

g:=sum(k*x^(k^2)/product(1-x^(2*i),i =1..k),k=1..20):gser:=series(g,x=0,52): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, Feb 14 2006

max = 100; s = Sum[ k*x^(k^2) / Product[1-x^(2*j), {j, 1, k}], {k, 1, Sqrt[max] // Ceiling}]; CoefficientList[ Series[s, {x, 0, max}], x] (* Jean-François Alcover, Feb 19 2015, after Vladeta Jovovic *)

N=66;  S=2+sqrtint(N); x='x+O('x^N);
gf=sum(n=0, S, n*x^(n^2)/prod(k=1,n, 1-x^(2*k)) );
concat( [0], Vec(gf) )
\\ Joerg Arndt, Feb 18 2014

G.f.: (Sum_{k>=1} x^(2*k-1)/(1 + x^(2*k-1))) * Product_{m>=1} (1 + x^(2m-1)).
G.f.: Sum_{k>=1} k*x^(k^2)/Product_{j=1..k} (1 - x^(2*j)). - Vladeta Jovovic, Aug 06 2004
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/6)) / (Pi * 2^(5/4) * n^(1/4)). - Vaclav Kotesovec, May 20 2018

A129838 Number of up/down (or down/up) compositions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 18, 21, 30, 38, 52, 78, 97, 128, 170, 222, 285, 421, 510, 683, 872, 1148, 1440, 1893, 2576, 3209, 4151, 5313, 6784, 8615, 10969, 13755, 18573, 22713, 29173, 36536, 46705, 57899, 73696, 91076, 114777, 148531, 182813, 228938, 287042

Offset: 0

Vladeta Jovovic, May 21 2007

Original name was: Number of alternating compositions of n into distinct parts.

A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. - Gus Wiseman, Jan 15 2022

			From _Gus Wiseman_, Jan 15 2022: (Start)
The a(1) = 1 through a(8) = 8 up/down strict compositions (non-strict A025048):
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
                          (2,3)  (2,4)    (2,5)    (2,6)
                                 (1,3,2)  (3,4)    (3,5)
                                 (2,3,1)  (1,4,2)  (1,4,3)
                                          (2,4,1)  (1,5,2)
                                                   (2,5,1)
                                                   (3,4,1)
The a(1) = 1 through a(8) = 8 down/up strict compositions (non-strict A025049):
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)
                          (4,1)  (5,1)    (5,2)    (6,2)
                                 (2,1,3)  (6,1)    (7,1)
                                 (3,1,2)  (2,1,4)  (2,1,5)
                                          (4,1,2)  (3,1,4)
                                                   (4,1,3)
                                                   (5,1,2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

The case of permutations is A000111.
This is the up/down case of A032020.
This is the strict case of A129852/A129853, strong A025048/A025049.
The undirected version is A349054.
A001250 = alternating permutations, complement A348615.
A003242 = Carlitz compositions, complement A261983.
A011782 = compositions, unordered A000041.
A025047 = alternating compositions, complement A345192.
A349052 = weakly alternating compositions, complement A349053.
Cf. A003056, A008289, A008965, A015723, A072706, A128761, A218074, A345165, A345170, A345195, A349800.

g:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
    end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k)+b(n-k, k-1)))
    end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 22 2021

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@ Select[IntegerPartitions[n],UnsameQ@@#&],whkQ]],{n,0,15}] (* Gus Wiseman, Jan 15 2022 *)

G.f.: Sum_{k>=0} A000111(k)*x^(k*(k+1)/2)/Product_{i=1..k} (1-x^i). - Vladeta Jovovic, May 24 2007
a(n) = Sum_{k=0..A003056(n)} A000111(k) * A008289(n,k). - Alois P. Heinz, Dec 22 2021
a(n) = (A349054(n) + 1)/2. - Gus Wiseman, Jan 15 2022

a(0)=1 prepended by Alois P. Heinz, Dec 22 2021

Name changed from "alternating" to "up/down" by Gus Wiseman, Jan 15 2022

A238132 Number of parts in all partitions of n into even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 6, 6, 8, 12, 14, 18, 24, 32, 38, 50, 60, 76, 90, 110, 134, 162, 190, 228, 270, 322, 380, 446, 524, 616, 720, 838, 980, 1134, 1314, 1526, 1760, 2026, 2336, 2676, 3072, 3518, 4020, 4586, 5232, 5948, 6760, 7676, 8698, 9846, 11142, 12578

Offset: 0

Mircea Merca, Feb 18 2014

nonn

			a(8)=6 because the partitions of 8 into even number of distinct parts are: 7+1, 6+2 and 5+3.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function s_e(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A000005, A001318, A015723, A110654, A235963.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, (p->
      [p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
    end:
a:= n-> b(n$2)[3]:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 27 2015

max = 50; s = (1/2)*Product[1+x^k, {k, 1, max}]*Sum[x^k/(1+x^k), {k, 1, max}] - (1/2)*Product[1-x^k, {k, 1, max}]*Sum[x^k/(1-x^k), {k, 1, max}] + O[x]^(max+1); CoefficientList[s, x] (* Jean-François Alcover, Dec 27 2015 *)

a(n)=(1/2)*A015723(n)-(1/2)*sum{k=0..A235963(n)-1, (-1)^A110654(k)*A000005(n-A001318(k))}=A015723(n)-A238131(n).
G.f.: (1/2)*prod(k>=1, 1+x^k ) * sum(k>=1, x^k/(1+x^k) ) - (1/2)*prod(k>=1, 1-x^k) * sum(k>=1, x^k/(1-x^k) ).
G.f.: -(2 * (x; x)inf * (log(1-x) + psi_x(1)) + (-1; x)_inf * (log(1-x) + psi_x(1-log(-1)/log(x))))/(4*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - _Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 27 2018

A015724 Number of parts in all partitions of all the numbers in {1,2,...,n} into distinct parts.

Original entry on oeis.org

1, 2, 5, 8, 13, 21, 31, 44, 62, 87, 117, 157, 206, 269, 349, 447, 566, 715, 894, 1112, 1378, 1696, 2076, 2531, 3072, 3712, 4472, 5367, 6417, 7651, 9093, 10772, 12732, 15004, 17639, 20691, 24211, 28265, 32934, 38293, 44435, 51470, 59507, 68677

Offset: 1

Clark Kimberling

nonn

Partial sums of A015723.
G.f.: (1/(1 - x)) * Sum_{i>=1} x^i/(1 + x^i) * Product_{j>=1} (1 + x^j). - Ilya Gutkovskiy, Apr 04 2017
a(n) ~ 3^(3/4) * log(2) * exp(Pi*sqrt(n/3)) * n^(1/4) / Pi^2. - Vaclav Kotesovec, Jun 20 2025

Corrected by Vladeta Jovovic, Dec 03 2004

cp's OEIS Frontend

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A325505 Heinz number of the set of Heinz numbers of all strict integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A327622 Number A(n,k) of parts in all k-times partitions of n into distinct parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A079499 Total number of parts in all partitions of n into distinct odd parts.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula