A325537 -id:A325537 - cp's OEIS frontend

A015723 Number of parts in all partitions of n into distinct parts.

1, 1, 3, 3, 5, 8, 10, 13, 18, 25, 30, 40, 49, 63, 80, 98, 119, 149, 179, 218, 266, 318, 380, 455, 541, 640, 760, 895, 1050, 1234, 1442, 1679, 1960, 2272, 2635, 3052, 3520, 4054, 4669, 5359, 6142, 7035, 8037, 9170, 10460, 11896, 13517, 15349, 17394, 19691

Offset: 1

Clark Kimberling

nonn

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with a total of 1 + 2 + 2 + 3 = 8 parts, so a(6) = 8. - _Gus Wiseman_, May 09 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Arnold Knopfmacher, and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
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. See s(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.

Cf. A006128, A008289, A079499, A067619, A186545.
Column k=1 of: A210485, A213177, A327622.
Row lengths of A325537.
Cf. A022629, A066186, A066189, A325504, A325505, A325506, A325513, A325515.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013

nn=50; Rest[CoefficientList[Series[D[Product[1+y x^i,{i,1,nn}],y]/.y->1,{x,0,nn}],x]]  (* Geoffrey Critzer, Oct 29 2012; fixed by Vaclav Kotesovec, Apr 16 2016 *)
q[n_, k_] := q[n, k] = If[nVaclav Kotesovec, Apr 16 2016 *)
Table[Length[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,50}] (* Gus Wiseman, May 09 2019 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},
   Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

N=66;  q='q+O('q^N); gf=sum(n=0,N, n*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf) /* Joerg Arndt, Oct 20 2012 */

G.f.: sum(k>=1, x^k/(1+x^k) ) * prod(m>=1, 1+x^m ). Convolution of A048272 and A000009. - Vladeta Jovovic, Nov 26 2002
G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005
a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - Mircea Merca, Feb 26 2014
a(n) = Sum_{k>=1} k*A008289(n,k). - Vaclav Kotesovec, Apr 16 2016
G.f.: -(-1; x)inf * (log(1-x) + psi_x(1 - log(-1)/log(x)))/(2*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)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, May 19 2018
For n > 0, a(n) = A116676(n) + A116680(n). - Vaclav Kotesovec, May 26 2018

Extended and corrected by Naohiro Nomoto, Feb 24 2002

A066189 Sum of all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 6, 8, 15, 24, 35, 48, 72, 100, 132, 180, 234, 308, 405, 512, 646, 828, 1026, 1280, 1596, 1958, 2392, 2928, 3550, 4290, 5184, 6216, 7424, 8880, 10540, 12480, 14784, 17408, 20475, 24048, 28120, 32832, 38298, 44520, 51660, 59892, 69230, 79904

Offset: 0

Wouter Meeussen, Dec 15 2001

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with sum 6+5+1+4+2+3+2+1 = 24. - _Gus Wiseman_, May 09 2019

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

Row sums of A026793, A118457, A246688, A325537.

Cf. A015723, A022629, A066186, A147655, A325504, A325505, A325506, A325513, A325515, A325537.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>n, [0$2],
      b(n, i+1)+(p-> p+[0, i*p[1]])(b(n-i, i+1))))
    end:
a:= n-> b(n, 1)[2]:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 01 2014

PartitionsQ[ Range[ 60 ] ]Range[ 60 ]
nmax=60; CoefficientList[Series[x*D[Product[1+x^k, {k, 1, nmax}], x], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)

G.f.: sum(n>=1, n*q^(n-1)/(1+q^n) ) * prod(n>=1, 1+q^n ). - Joerg Arndt, Aug 03 2011
a(n) = n * A000009(n). - Vaclav Kotesovec, Sep 25 2016
G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^k). - Vaclav Kotesovec, Nov 21 2016
a(n) = A056239(A325506(n)). - Gus Wiseman, May 09 2019

A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).

Original entry on oeis.org

1, 2, 3, 11, 17, 40, 86, 153, 283, 547, 1069, 1737, 3238, 5340, 9574, 17251, 27897, 45845, 78601, 126725, 207153, 353435, 550422, 881454, 1393870, 2239938, 3473133, 5546789, 8762663, 13341967, 20676253, 31774563, 48248485, 74174759, 111904363, 170184798

Offset: 0

Neil Fernandez, Nov 09 2008

nonn

Sum of all squarefree numbers whose prime indices sum to n. 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. - Gus Wiseman, May 09 2019

			Form a product from the primes: (1 + 2*x) * (1 + 3*x^2) * (1 + 5*x^3) * ...* (1 + prime(n)*x^n) * ... Multiplying out gives 1 + 2*x + 3*x^2 + 11*x^3 + ..., so the sequence begins 1, 2, 3, 11, ....
From _Petros Hadjicostas_, Apr 10 2020: (Start)
Let f(m) = prime(m). Using the strict partitions of n (see A000009), we get:
a(1) = f(1) = 2,
a(2) = f(2) = 3,
a(3) = f(3) + f(1)*f(2) = 5 + 2*3 = 11,
a(4) = f(4) + f(1)*f(3) = 7 + 2*5 = 17,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 11 + 2*7 + 3*5 = 40,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 13 + 2*11 + 3*7 + 2*3*5 = 86,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 17 + 2*13 + 3*11 + 5*7 + 2*3*7 = 153. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..9000 (first 1001 terms from Harvey P. Dale)

Row sums of A246867 and A258323.

Cf. A000009,A005117, A015723, A022629, A056239, A066189, A112798, A145519, A147541, A325504, A325506, A325537.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 05 2014

nn=40;Take[Rest[CoefficientList[Expand[Times@@Table[1+Prime[n]x^n,{n,nn}]],x]],nn] (* Harvey P. Dale, Jul 01 2012 *)

a(n) = [x^n] Product_{k>=1} 1+prime(k)*x^k. - Alois P. Heinz, Sep 05 2014

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = prime(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 10 2020

More terms from Harvey P. Dale, Jul 01 2012
a(0)=1 inserted by Alois P. Heinz, Sep 05 2014
Name edited by Petros Hadjicostas, Apr 10 2020

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A015723 Number of parts in all partitions of n into distinct parts.

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A066189 Sum of all partitions of n into distinct parts.

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A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).

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