A325765 -id:A325765 - cp's OEIS frontend

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

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

A325770 Number of distinct nonempty contiguous subsequences of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 20 2019

nonn

After a(1) = 0, first differs from A305611 at a(42) = 6, A305611(42) = 7.

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

			The a(84) = 9 distinct nonempty contiguous subsequences of (4,2,1,1) are (1), (2), (4), (1,1), (2,1), (4,2), (2,1,1), (4,2,1), (4,2,1,1).

Cf. A002865, A103295, A112798, A124771, A276024, A325765, A325768, A325769, A335519, A335838.

Table[Length[Union[ReplaceList[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]],{_,s__,_}:>{s}]]],{n,30}]

a(n) = A335519(n) - 1.

Name corrected by Gus Wiseman, Jun 27 2020

A325763 Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 280, 288, 300, 320, 324, 336, 352, 360, 384, 392, 400, 416, 432, 448, 480, 486, 500, 504, 512

Offset: 1

Gus Wiseman, May 19 2019

nonn

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

The enumeration of these partitions by sum appears to be A002865.

			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}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    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}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}

Cf. A002033, A103295, A103300, A169942, A325676, A325685, A325764, A325765.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]]&]

A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 12, 14, 14, 15, 15, 18, 17, 19, 18, 20, 20, 22, 21, 25, 23, 26, 25, 28, 26, 29, 27, 31, 29, 32, 31, 34, 33, 35, 34, 38, 35, 41, 37, 42, 40, 43, 41, 45, 42, 46, 44, 48, 45, 50, 46, 52, 49, 53

Offset: 0

Gus Wiseman, Oct 13 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			Representative partitions for each of the a(16) = 11 alternating products:
         (16) -> 16
     (14,1,1) -> 14
     (12,2,2) -> 12
     (10,3,3) -> 10
      (8,4,4) -> 8
  (9,3,2,1,1) -> 6
     (10,4,2) -> 5
     (12,3,1) -> 4
  (6,4,2,2,2) -> 3
     (10,5,1) -> 2
        (8,8) -> 1

The even-length version is A000035.
The non-reverse version is A028310.
The version for factorizations has special cases:
- no changes: A046951
- non-reverse: A046951
- non-integer: A038548
- odd-length: A046951 + A010052
- non-reverse non-integer: A347460
- non-integer odd-length: A347708
- non-reverse odd-length: A046951 + A010052
- non-reverse non-integer odd-length: A347708
The odd-length version is a(n) - A059841(n).
These partitions are counted by A347445, non-reverse A347446.
Counting non-integers gives A347462, non-reverse A347461.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A119620 counts partitions with alternating product 1, ranked by A028982.
A276024 counts distinct positive subset-sums of partitions, strict A284640.
A304792 counts distinct subset-sums of partitions.
A325534 counts separable partitions, complement A325535.
A345926 counts possible alternating sums of permutations of prime indices.
Cf. A000070, A002033, A002219, A108917, A122768, A325765, A344654, A344740, A347443, A347444, A347448, A347449.

revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[Union[revaltprod/@IntegerPartitions[n]],IntegerQ]],{n,0,30}]

A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 54, 56, 64, 100, 128, 162, 176, 256, 392, 416, 486, 500, 512, 1024, 1088, 1458, 1936, 2048, 2432, 2500, 2744, 4096, 4374, 5408, 5888, 8192, 12500, 13122, 14848, 16384, 18496, 19208, 21296, 31744, 32768, 39366, 46208, 62500, 65536

Offset: 1

Gus Wiseman, May 20 2019

nonn

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

The enumeration of these partitions by sum is given by A325765.

			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}
    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}
   256: {1,1,1,1,1,1,1,1}
   392: {1,1,1,4,4}
   416: {1,1,1,1,1,6}
   486: {1,2,2,2,2,2}
   500: {1,1,3,3,3}
   512: {1,1,1,1,1,1,1,1,1}

Cf. A002033, A056239, A103295, A103300, A112798, A143823, A169942, A325676, A325685, A325763, A325765, A325769, A325770.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],UnsameQ@@Total/@Union[ReplaceList[primeMS[#],{_,s__,_}:>{s}]]&&Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#],{_,s__,_}:>Plus[s]]]&]

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

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A325770 Number of distinct nonempty contiguous subsequences of the integer partition with Heinz number n.

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A325763 Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.

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A347707 Number of distinct possible integer reverse-alternating products of integer partitions of n.

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A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

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