A178830 -id:A178830 - cp's OEIS frontend

A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.

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

Offset: 0

Gus Wiseman, Jul 07 2019

nonn

Essentially the same as A178830. - R. J. Mathar, Jul 12 2019

			The initial terms count the following subsets:
   1: {1}
   3: {1,2}
   5: {3,5}
   6: {3,4,5}
   7: {2,4,5,7}
   8: {2,4,5,6,7}
   9: {2,3,5,6,7,9}
  10: {4,5,6,8,9,10}
  10: {2,3,5,6,7,8,9}
  10: {1,2,3,4,5,8,9,10}
Also the number of subsets of {1..n} whose product is equal to the sum of their complement. For example, the initial terms count the following subsets:
   1: {}
   3: {3}
   5: {1,2,4}
   6: {1,2,6}
   7: {1,3,6}
   8: {1,3,8}
   9: {1,4,8}
  10: {6,7}
  10: {1,4,10}
  10: {1,2,3,7}

Giovanni Resta, Table of n, a(n) for n = 0..500

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325044, A325538, A326172, A326173, A326174, A326175, A326179, A326180.

b:= proc(n, s, p)
      `if`(s=p, 1, `if`(n<1, 0, b(n-1, s, p)+
      `if`(s-n b(n, n*(n+1)/2, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 12 2019

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

a(21)-a(83) from Giovanni Resta, Jul 08 2019

A207852 Smallest number m such that there are exactly n ways to partition the numbers {1,...,m} into nonempty sets P and S with the product of the elements of P equal to the sum of elements in S.

Original entry on oeis.org

1, 3, 12, 10, 19, 26, 33, 39, 55, 74, 48, 62, 71, 99, 45, 140, 96, 176, 104, 144, 159, 175, 230, 191, 320, 328, 240, 334, 259, 344, 279, 308, 303, 505, 419, 560, 714, 550, 455, 665, 684, 670, 751, 935, 899, 800, 1051, 776, 928, 602, 749, 1104, 689, 1295, 1364

Offset: 0

Reinhard Zumkeller, Feb 21 2012

nonn

A178830(a(n)) = n and A178830(m) <> n for m < a(n).

			a(1) =  3: 3 = 1+2;
a(2) = 12: 1*5*12 = 2+3+4+6+7+8+9+10+11, 2*4*8 = 1+3+5+6+7+9+10+11+12;
a(3) = 10: 1*2*3*7 = 4+5+6+8+9+10, 1*4*10 = 2+3+5+6+7+8+9, 6*7 = 1+2+3+4+5+8+9+10.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a207852 n = (fromJust $ elemIndex n a178830_list) + 1

a(25)-a(54) from Alois P. Heinz, Jun 07 2012

A213237 Number of distinct values v satisfying v = sum of elements in S = product of elements in P for any partition of {1,...,n} into two sets S and P.

Original entry on oeis.org

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

Offset: 1

Alois P. Heinz, Jun 07 2012

nonn

			a(1) = 1: S={1}, P={}, v=1.
a(2) = 0: no partition of {1,2} satisfies the condition.
a(3) = 1: S={1,2}, P={3}, v=3.
a(10) = 2: three partitions of {1,2,...,10} into S and P satisfy v = Sum_{i:S} i = Product_{k:P} k but there are only two distinct values v: S={2,3,5,6,7,8,9}, P={1,4,10}, v=40; S={4,5,6,8,9,10}, P={1,2,3,7}, v=42; S={1,2,3,4,5,8,9,10}, P={6,7}, v=42.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

The values v are in A213238.

Cf. A178830, A207852.

b:= proc(n, s, p)
      `if`(s=p, {s}, `if`(n<1, {}, {b(n-1, s, p)[],
      `if`(s-n nops(b(n, n*(n+1)/2, 1)):
seq(a(n), n=1..100);

b[n_, s_, p_] := b[n, s, p] = If[s == p, {s}, If[n < 1, {}, Union[b[n - 1, s, p], If[s - n < p n, {}, b[n - 1, s - n, p n]]]]];
a[n_] := Length[b[n, n(n+1)/2, 1]];
Array[a, 100] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 07 2019

nonn

Also by definition the number of subsets whose sum is one fewer than the product of their complement.

			The initial terms count the following subsets:
   0: {}
   1: {1}
   2: {2}
   3: {1,3}
   4: {2,3}
   7: {4,5}
  10: {1,6,7}
  12: {7,9}
  12: {1,2,4,8}
  14: {2,5,9}
  14: {1,2,4,11}
  15: {1,3,5,7}
  16: {3,4,10}
  16: {1,3,5,8}
  17: {1,10,13}
  18: {2,5,15}
  19: {11,15}
  19: {1,2,6,14}
  20: {1,4,6,8}

Giovanni Resta, Table of n, a(n) for n = 0..2500

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325041, A326172, A326173, A326174, A326175, A326179, A326180, A326441.

Table[Length[Select[Subsets[Range[n]],1+Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]
ric[n_, pr_, s_, lst_, t_] := Block[{k}, If[pr == t-s, cnt++]; Do[ If[pr k <= t, ric[n, pr k, s + k, k, t], Break[]], {k, lst+1, n}]]; a[n_] := (cnt = 0; ric[n, 1, 0, 0, n (n + 1)/2 + 1]; cnt); a /@ Range[0, 85] (* Giovanni Resta, Sep 13 2019 *)

More terms from Alois P. Heinz, Jul 12 2019

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A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.

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A207852 Smallest number m such that there are exactly n ways to partition the numbers {1,...,m} into nonempty sets P and S with the product of the elements of P equal to the sum of elements in S.

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A213237 Number of distinct values v satisfying v = sum of elements in S = product of elements in P for any partition of {1,...,n} into two sets S and P.

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Maple

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A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

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Mathematica

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