A126796 -id:A126796 - cp's OEIS frontend

A002033 Number of perfect partitions of n.

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 44, 1, 3, 8, 32, 3, 13, 1, 8, 3, 13, 1, 76, 1, 3, 8, 8, 3, 13, 1, 48, 8, 3, 1, 44, 3, 3, 3, 20, 1, 44, 3, 8, 3, 3, 3, 112

Offset: 0

N. J. A. Sloane

A perfect partition of n is one which contains just one partition of every number less than n when repeated parts are regarded as indistinguishable. Thus 1^n is a perfect partition for every n; and for n = 7, 4 1^3, 4 2 1, 2^3 1 and 1^7 are all perfect partitions. [Riordan]
Also number of ordered factorizations of n+1, see A074206.
Also number of gozinta chains from 1 to n (see A034776). - David W. Wilson
a(n) is the permanent of the n X n matrix with (i,j) entry = 1 if j|i+1 and = 0 otherwise. For n=3 the matrix is {{1, 1, 0}, {1, 0, 1}, {1, 1, 0}} with permanent = 2. - David Callan, Oct 19 2005
Appears to be the number of permutations that contribute to the determinant that gives the Moebius function. Verified up to a(9). - Mats Granvik, Sep 13 2008
Dirichlet inverse of A153881 (assuming offset 1). - Mats Granvik, Jan 03 2009
Equals row sums of triangle A176917. - Gary W. Adamson, Apr 28 2010
A partition is perfect iff it is complete (A126796) and knapsack (A108917). - Gus Wiseman, Jun 22 2016
a(n) is also the number of series-reduced planted achiral trees with n + 1 unlabeled leaves, where a rooted tree is series-reduced if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. Also Moebius transform of A067824. - Gus Wiseman, Jul 13 2018

			n=0: 1 (the empty partition)
n=1: 1 (1)
n=2: 1 (11)
n=3: 2 (21, 111)
n=4: 1 (1111)
n=5: 3 (311, 221, 11111)
n=6: 1 (111111)
n=7: 4 (4111, 421, 2221, 1111111)
From _Gus Wiseman_, Jul 13 2018: (Start)
The a(11) = 8 series-reduced planted achiral trees with 12 unlabeled leaves:
  (oooooooooooo)
  ((oooooo)(oooooo))
  ((oooo)(oooo)(oooo))
  ((ooo)(ooo)(ooo)(ooo))
  ((oo)(oo)(oo)(oo)(oo)(oo))
  (((ooo)(ooo))((ooo)(ooo)))
  (((oo)(oo)(oo))((oo)(oo)(oo)))
  (((oo)(oo))((oo)(oo))((oo)(oo)))
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.
D. E. Knuth, The Art of Computer Programming, Pre-Fasc. 3b, Sect. 7.2.1.5, no. 67, p. 25.
P. A. MacMahon, The theory of perfect partitions and the compositions of multipartite numbers, Messenger Math., 20 (1891), 103-119.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 123-124.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..9999
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
HoKyu Lee, Double perfect partitions, Discrete Math., 306 (2006), 519-525.
Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1970
Eric Weisstein's World of Mathematics, Perfect Partition
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Index entries for "core" sequences

Same as A074206, up to the offset and initial term there.
Cf. A001055, A050324.
a(A002110)=A000670.
Cf. A000123, A100529, A117621.
Cf. A176917.
For parity see A008966.
Cf. A126796, A108917.
Cf. A001678, A003238, A067824, A167865, A214577, A289078, A292504, A316782.

a := array(1..150): for k from 1 to 150 do a[k] := 0 od: a[1] := 1: for j from 2 to 150 do for m from 1 to j-1 do if j mod m = 0 then a[j] := a[j]+a[m] fi: od: od: for k from 1 to 150 do printf(`%d,`,a[k]) od: # James Sellers, Dec 07 2000
# alternative
A002033 := proc(n)
    option remember;
    local a;
    if n <= 2 then
        return 1;
    else
        a := 0 ;
        for i from 0 to n-1 do
            if modp(n+1,i+1) = 0 then
                a := a+procname(i);
            end if;
        end do:
    end if;
    a ;
end proc: # R. J. Mathar, May 25 2017

a[0] = 1; a[1] = 1; a[n_] := a[n] = a /@ Most[Divisors[n]] // Total; a /@ Range[96]  (* Jean-François Alcover, Apr 06 2011, updated Sep 23 2014. NOTE: This produces A074206(n) = a(n-1). - M. F. Hasler, Oct 12 2018 *)

A002033(n) = if(n,sumdiv(n+1,i,if(i<=n,A002033(i-1))),1) \\ Michael B. Porter, Nov 01 2009, corrected by M. F. Hasler, Oct 12 2018

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A002033(n):
    if n <= 1:
        return 1
    return sum(A002033(i-1) for i in divisors(n+1,generator=True) if i <= n) # Chai Wah Wu, Jan 12 2022

From David Wasserman, Nov 14 2006: (Start)
a(n-1) = Sum_{i|d, i < n} a(i-1).
a(p^k-1) = 2^(k-1).
a(n-1) = A067824(n)/2 for n > 1.
a(A122408(n)-1) = A122408(n)/2. (End)
a(A025487(n)-1) = A050324(n). - R. J. Mathar, May 26 2017
a(n) = (A253249(n+1)+1)/4, n > 0. - Geoffrey Critzer, Aug 19 2020

Edited by M. F. Hasler, Oct 12 2018

A325781 Heinz numbers of complete integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 160, 162, 168, 176, 180, 192, 198, 200, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288, 294, 300

Offset: 1

Gus Wiseman, May 21 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sum of prime indices of n is A056239(n). A number is in this sequence iff its divisors have sums of prime indices covering an initial interval of nonnegative integers. For example, the divisors of 60 are {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, with respective sums of prime indices {0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7}, so 60 is in the sequence.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    30: {1,2,3}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    42: {1,2,4}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    64: {1,1,1,1,1,1}

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A325684, A325770, A325780, A325782.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],normQ[hwt/@Rest[Divisors[#]]]&]

A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Naohiro Nomoto, Jan 07 2004

This function depends only on the prime signature of n. - Franklin T. Adams-Watters, Mar 10 2006
a(n) is the number of perfect divisors of n. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) > 1 for perfect powers n = A001597(m) for m > 2. - Jaroslav Krizek, Jan 23 2010
Also the number of uniform perfect integer partitions of n - 1. An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset. The Heinz numbers of these partitions are given by A326037. The a(16) = 3 partitions are: (8,4,2,1), (4,4,4,1,1,1), (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1). - Gus Wiseman, Jun 07 2019
The record values occur at 1 and at 2^A002182(n) for n > 1. - Amiram Eldar, Nov 06 2020

			144 = 2^4 * 3^2, gcd(4,2) = 2, d(2) = 2, so a(144) = 2. The representations are 144^1 and 12^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:          1, 2, 3, 4, 5, 6, 7, 8, 9, ...
----------------------------------------------------
1st powers: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012)
Squares:    1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:      1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
Quartics:   1, 0, 0, 0, 0, 0, 0, 0, 0, ... (A374016)
...
Sum:       oo, 1, 1, 2, 1, 1, 1, 2, 2, ...
a(1)=1:     1, 1, 1, 2, 1, 1, 1, 2, 2, ... (= this sequence). (End)

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, A new arithmetic function of combinatorial significance, J. Number Theory, Vol. 5, No. 3 (1973), pp. 218-223. 1973JNT.....5..218G
Jan Mycielski, Sur les représentations des nombres naturels par des puissances à base et exposant naturels, Colloquium Mathematicum, Vol. 2 (1951), pp. 254-260. See gamma(n).
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Cf. A000005, A277564, A278028.
Cf. A000961, A002033, A002182, A007916, A047966, A070941, A108917, A126796, A276024, A326035, A326036, A326037.
Cf. A010052, A010057, A374016, A259362.

with(numtheory):
A089723 := proc(n) local t1,t2,g,j;
if n=1 then 1 else
t1:=ifactors(n)[2]; t2:=nops(t1); g := t1[1][2];
for j from 2 to t2 do g:=gcd(g,t1[j][2]); od:
tau(g); fi; end;
[seq(A089723(n),n=1..100)]; # N. J. A. Sloane, Nov 10 2016

Table[DivisorSigma[0, GCD @@ FactorInteger[n][[All, 2]]], {n, 100}] (* Gus Wiseman, Jun 12 2017 *)

a(n) = if (n==1, 1, numdiv(gcd(factor(n)[,2]))); \\ Michel Marcus, Jun 13 2017

from math import gcd
from sympy import factorint, divisor_sigma
def a(n):
    if n == 1: return 1
    e = list(factorint(n).values())
    g = e[0]
    for ei in e[1:]: g = gcd(g, ei)
    return divisor_sigma(g, 0)
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 15 2021

If n = Product p_i^e_i, a(n) = d(gcd()). - Franklin T. Adams-Watters, Mar 10 2006
Sum_{n=1..m} a(n) = A255165(m) + 1. - Richard R. Forberg, Feb 16 2015
Sum_{n>=2} a(n)/n^s = Sum_{n>=2} 1/(n^s-1) = Sum_{k>=1} (zeta(s*k)-1) for all real s with Re(s) > 1 (Golomb, 1973). - Amiram Eldar, Nov 06 2020
For n > 1, a(n) = Sum_{i=1..floor(n/2)} floor(n^(1/i))-floor((n-1)^(1/i)). - Wesley Ivan Hurt, Dec 08 2020
Sum_{n>=1} (a(n)-1)/n = 1 (Mycielski, 1951). - Amiram Eldar, Jul 15 2021
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = 1 + A259362(n) = 1 + A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
G.f.: x + Sum_{j>=2, k>=1} x^(j^k). (End)

A188431 The number of n-full sets, F(n).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 2, 1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 10, 13, 14, 17, 20, 25, 28, 34, 40, 46, 54, 62, 69, 80, 90, 102, 115, 131, 144, 167, 186, 213, 239, 273, 304, 349, 388, 441, 495, 563, 625, 710, 790, 890, 990, 1114, 1232, 1387, 1530, 1713, 1894, 2119, 2330, 2605, 2866, 3192, 3512, 3910, 4289, 4774, 5237, 5809, 6377, 7068, 7739

Offset: 0

Madjid Mirzavaziri, Mar 31 2011

nonn

Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. Then F(n) denotes the number of n-full sets.
Also the number of distinct and complete partitions of n, by definition, which are counted by A000009 and A126796. - George Beck, Nov 06 2017
An integer partition of n is complete (see also A325781) if every number from 0 to n is the sum of some submultiset of the parts. The Heinz numbers of these partitions are given by A325986. - Gus Wiseman, May 31 2019

			a(26) = 10, because there are 10 26-full sets: {1,2,4,5,6,8}, {1,2,3,5,7,8}, {1,2,3,5,6,9}, {1,2,3,4,7,9}, {1,2,3,4,6,10}, {1,2,3,4,5,11}, {1,2,4,8,11}, {1,2,4,7,12}, {1,2,4,6,13}, {1,2,3,7,13}.
G.f.: 1 = 1/(1+x) + 1*x/((1+x)*(1+x^2)) + 0*x^2/((1+x)*(1+x^2)*(1+x^3)) + 1*x^3/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...+ a(n)*x^n / Product_{k=1..n+1} (1+x^k) +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..1000 from Reinhard Zumkeller)
Mohammad Saleh Dinparvar, Python program
L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
Arash Sal Moslehian, JavaScript p5 program for three different algorithms

Cf. A188429, A188430, A126796.

Cf. A002033, A103295, A108917, A276024, A325763, A325765, A325781, A325782, A325788, A325986, A325790, A325791.

import Data.MemoCombinators (memo2, integral, Memo)
a188431 n = a188431_list !! (n-1)
a188431_list = map
   (\x -> sum [fMemo x i | i <- [a188429 x .. a188430 x]]) [1..] where
   fMemo = memo2 integral integral f
   f _ 1 = 1
   f m i = sum [fMemo (m - i) j |
                j <- [a188429 (m - i) .. min (a188430 (m - i)) (i - 1)]]
-- Reinhard Zumkeller, Aug 06 2015

sums:= proc(s) local i, m;
          m:= max(s[]);
         `if`(m<1, {}, {m, seq([i, i+m][], i=sums(s minus {m}))})
       end:
a:= proc(n) local b;
      b:= proc(i,s) local si;
            if i=1 then `if`(sums(s)={$1..n}, 1, 0)
          else si:= s union {i};
               b(i-1, s)+ `if`(max(sums(si)[])>n, 0, b(i-1, si))
            fi
          end; b(n, {1})
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 03 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2n or i>n-i+1, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 20 2017

Sums[s_] := Sums[s] = With[{m = Max[s]}, If[m < 1, {}, Union @ Flatten @ Join[{m}, Table[{i, i + m}, {i, Sums[s ~Complement~ {m}]}]]]];
a[n_] := Module[{b}, b[i_, s_] := b[i, s] = Module[{si}, If[i == 1, If[Sums[s] == Range[n], 1, 0], si = s ~Union~ {i}; b[i-1, s] + If[Max[ Sums[si]] > n, 0, b[i - 1, si]]]]; b[n, {1}]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 80}] (* Jean-François Alcover, Apr 12 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Union[Total/@Union[Subsets[#]]]==Range[0,n]&]],{n,30}] (* Gus Wiseman, May 31 2019 *)

/* As coefficients in g.f. */
{a(n)=local(A=[1]); for(i=1, n+1, A=concat(A,0); A[#A]=polcoeff(1 - sum(m=1,#A,A[m]*x^m/prod(k=1, m, 1+x^k +x*O(x^#A) )), #A) ); A[n+1]}
for(n=0, 50, print1(a(n),", ")) /* Paul D. Hanna, Mar 06 2012 */

F(n) = Sum_(i=L(n) .. U(n), F(n,i)), where F(n,i) = Sum_(j=L(n-i) .. min(U(n-i),i-1), F(n-i,j) ) and L(n), U(n) are defined in A188429 and A188430, respectively.
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+x^k), with a(0)=1. - Paul D. Hanna, Mar 08 2012
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(3/4), where c = 0.03316508... - Vaclav Kotesovec, Oct 21 2019

More terms from Alois P. Heinz, Apr 03 2011

a(0)=1 prepended by Alois P. Heinz, May 20 2017

A103295 Number of complete rulers with length n.

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 17, 33, 63, 128, 248, 495, 988, 1969, 3911, 7857, 15635, 31304, 62732, 125501, 250793, 503203, 1006339, 2014992, 4035985, 8080448, 16169267, 32397761, 64826967, 129774838, 259822143, 520063531, 1040616486, 2083345793, 4168640894, 8342197304, 16694070805, 33404706520, 66832674546, 133736345590

Offset: 0

Peter Luschny, Feb 28 2005

nonn

For definitions, references and links related to complete rulers see A103294.
Also the number of compositions of n whose consecutive subsequence-sums cover an initial interval of the positive integers. For example, (2,3,1) is such a composition because (1), (2), (3), (3,1), (2,3), and (2,3,1) are subsequences with sums covering {1..6}. - Gus Wiseman, May 17 2019
a(n) ~ c*2^n, where 0.2427 < c < 0.2459. - Fei Peng, Oct 17 2019

			a(4) = 4 counts the complete rulers with length 4, {[0,2,3,4],[0,1,3,4],[0,1,2,4],[0,1,2,3,4]}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..49
Scott Harvey-Arnold, Steven J. Miller, and Fei Peng, Distribution of missing differences in diffsets, arXiv:2001.08931 [math.CO], 2020.
Peter Luschny, Perfect rulers
Hugo Pfoertner, Count complete rulers of given length. FORTRAN program.
Index entries for sequences related to perfect rulers.
Gus Wiseman, Illustration of A103295.

Cf. A103300 (Perfect rulers with length n). Main diagonal of A349976.

Cf. A000079, A126796, A143823, A169942, A325676, A325677, A325683, A325684, A325685.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SubsetQ[ReplaceList[#,{_,s__,_}:>Plus[s]],Range[n]]&]],{n,0,15}] (* Gus Wiseman, May 17 2019 *)

a(n) = Sum_{i=0..n} A103294(n, i) = Sum_{i=A103298(n)..n} A103294(n, i).

a(30)-a(36) from Hugo Pfoertner, Mar 17 2005
a(37)-a(38) from Hugo Pfoertner, Dec 10 2021
a(39) from Hugo Pfoertner, Dec 16 2021

A326020 Number of complete subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 15, 27, 50, 95, 185, 365, 724, 1441, 2873, 5735, 11458, 22902, 45789, 91561, 183102, 366180, 732331, 1464626, 2929209, 5858367, 11716674, 23433277, 46866473, 93732852, 187465596, 374931067, 749861989, 1499723808, 2999447418

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,3,4}
                                      {1,2,3,5}    {1,2,3,5}
                                      {1,2,4,5}    {1,2,3,6}
                                      {1,2,3,4,5}  {1,2,4,5}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,4,5,6}
                                                   {1,2,3,4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326021, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,0,10}]

a(17)-a(34) from Charlie Neder, Jun 05 2019

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Gus Wiseman, Sep 16 2023

Conjecture: Positions of strictly positive rows are given by A048166.

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71

Robert Price, Table of n, a(n) for n = 1..300

Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]

A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

The complement (complete partitions) is A126796.

			The a(0) = 0 through a(8) = 12 partitions:
  .  .  (2)  (3)  (4)    (5)    (6)      (7)      (8)
                  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)
                  (3,1)  (4,1)  (4,2)    (5,2)    (5,3)
                                (5,1)    (6,1)    (6,2)
                                (2,2,2)  (3,2,2)  (7,1)
                                (4,1,1)  (3,3,1)  (3,3,2)
                                         (5,1,1)  (4,2,2)
                                                  (4,3,1)
                                                  (5,2,1)
                                                  (6,1,1)
                                                  (2,2,2,2)
                                                  (5,1,1,1)

Joerg Arndt, Table of n, a(n) for n = 0..10000

For parts instead of sums we have A047967/A365919, ranks A080259/A055932.
The complement is A126796, ranks A325781, strict A188431.
These partitions have ranks A365830.
The strict case is A365831.
Row sums of A365923 without the first column, strict A365545.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]

a(n) = A000041(n) - A126796(n).

A325780 Heinz numbers of perfect integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878

Offset: 1

Gus Wiseman, May 21 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     16: {1,1,1,1}
     18: {1,2,2}
     20: {1,1,3}
     32: {1,1,1,1,1}
     42: {1,2,4}
     54: {1,2,2,2}
     56: {1,1,1,4}
     64: {1,1,1,1,1,1}
    100: {1,1,3,3}
    128: {1,1,1,1,1,1,1}
    162: {1,2,2,2,2}
    176: {1,1,1,1,5}
    234: {1,2,2,6}
    256: {1,1,1,1,1,1,1,1}
    260: {1,1,3,6}

Equals the sorted concatenation of the triangle A258119.
A subsequence of A299702 and A325781.
Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]

Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).

A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 6, 8, 9, 11, 13, 16, 21, 25, 31, 36, 43, 50, 59, 69, 82, 96, 113, 131, 155, 179, 208, 239, 276, 315, 362, 414, 472, 539, 614, 698, 795, 902, 1023, 1158, 1311, 1479, 1672, 1881, 2118, 2377, 2671, 2991, 3354, 3748, 4194, 4679, 5223, 5815

Offset: 0

Gus Wiseman, Sep 28 2023

nonn

			The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).
The a(2) = 1 through a(10) = 9 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (10)
            (3,1)  (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)
                   (4,1)  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)
                                 (6,1)  (7,1)    (7,2)    (8,2)
                                        (4,3,1)  (8,1)    (9,1)
                                        (5,2,1)  (4,3,2)  (5,3,2)
                                                 (5,3,1)  (5,4,1)
                                                 (6,2,1)  (6,3,1)
                                                          (7,2,1)

For parts instead of sums we have ranks A080259, A055932.
The strict complement is A188431, non-strict A126796 (ranks A325781).
Row sums of A365545 without the first column, non-strict A365923.
The non-strict version is A365924, ranks A365830.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A006827, A047967, A299701, A304792, A364350, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[nmz[#]]>0&]],{n,0,15}]

cp's OEIS Frontend

A002033 Number of perfect partitions of n.

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A325781 Heinz numbers of complete integer partitions.

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A089723 a(1)=1; for n>1, a(n) gives number of ways to write n as n = x^y, 2 <= x, 1 <= y.

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A188431 The number of n-full sets, F(n).

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A103295 Number of complete rulers with length n.

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A326020 Number of complete subsets of {1..n}.

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A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

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