A304796 -id:A304796 - cp's OEIS frontend

A304795 Number of positive special sums of the integer partition with Heinz number n.

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

Offset: 1

Gus Wiseman, May 18 2018

nonn

A positive special sum of y is a number n > 0 such that exactly one submultiset of y sums to n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(36) = 4 special sums are 1, 3, 5, 6, corresponding to the submultisets (1), (21), (221), (2211), with Heinz numbers 2, 6, 18, 36.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000712, A056239, A108917, A122768, A276024, A284640, A296150, A299701, A299702, A301854, A301855, A301957, A304793, A304796.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
Table[Length[uqsubs[primeMS[n]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A304795(n) = { my(m=Map(),s,k=0,c); fordiv(n,d,if(!mapisdefined(m,s = v056239[d],&c), mapput(m,s,1), mapput(m,s,c+1))); sumdiv(n,d,(1==mapget(m,v056239[d])))-1; }; \\ Antti Karttunen, Jul 02 2018

More terms from Antti Karttunen, Jul 02 2018

A299764 Number of special products of factorizations of n into factors > 1.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 18, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 32, 6, 6, 6, 44, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 72, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 51, 6, 22, 2, 16, 6, 22, 2, 104, 2, 6, 16, 16, 6

Offset: 1

Gus Wiseman, Jun 08 2018

nonn

A special product of a factorization f is a number n > 0 such that exactly one submultiset of f has product n.

			The a(12) = 16 special subset-products:
1<=(12), 12<=(12),
1<=(2*6), 2<=(2*6), 6<=(2*6), 12<=(2*6),
1<=(3*4), 3<=(3*4), 4<=(3*4), 12<=(3*4),
1<=(2*2*3), 2<=(2*2*3), 3<=(2*2*3), 4<=(2*2*3), 6<=(2*2*3), 12<=(2*2*3).
The a(16) = 18 special subset-products:
1<=(16), 16<=(16),
1<=(4*4), 4<=(4*4), 16<=(4*4),
1<=(2*8), 2<=(2*8), 8<=(2*8), 16<=(2*8),
1<=(2*2*4), 2<=(2*2*4), 8<=(2*2*4), 16<=(2*2*4),
1<=(2*2*2*2), 2<=(2*2*2*2), 4<=(2*2*2*2), 8<=(2*2*2*2), 16<=(2*2*2*2).

Cf. A001055, A292886, A299701, A301829, A301830, A301854, A301957, A301979, A304796.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sppr[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Times@@#&],Length[#]===1&];
Table[Length[Join@@sppr/@facs[n]],{n,30}]

A367108 Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 2, 3, 5, 7, 5, 4, 4, 5, 7, 11, 7, 6, 3, 6, 7, 11, 15, 11, 8, 7, 7, 8, 11, 15, 22, 15, 12, 10, 4, 10, 12, 15, 22, 30, 22, 16, 14, 12, 12, 14, 16, 22, 30, 42, 30, 22, 17, 17, 6, 17, 17, 22, 30, 42, 56, 42, 30, 25, 23, 20, 20, 23, 25, 30, 42, 56

Offset: 1

Gus Wiseman, Nov 18 2023

			Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   2   3   5
   7   5   4   4   5   7
  11   7   6   3   6   7  11
  15  11   8   7   7   8  11  15
  22  15  12  10   4  10  12  15  22
  30  22  16  14  12  12  14  16  22  30
  42  30  22  17  17   6  17  17  22  30  42
  56  42  30  25  23  20  20  23  25  30  42  56
  77  56  40  31  30  27   7  27  30  31  40  56  77
Row n = 5 counts the following partitions:
  (5)      (41)     (32)     (32)     (41)     (5)
  (41)     (311)    (311)    (311)    (311)    (41)
  (32)     (221)    (221)    (221)    (221)    (32)
  (311)    (2111)   (11111)  (11111)  (2111)   (311)
  (221)    (11111)                    (11111)  (221)
  (2111)                                       (2111)
  (11111)                                      (11111)
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (2211)    (411)     (411)     (51)
  (42)      (321)     (321)     (111111)  (321)     (321)     (42)
  (411)     (3111)    (3111)              (3111)    (3111)    (411)
  (33)      (2211)    (222)               (222)     (2211)    (33)
  (321)     (21111)   (111111)            (111111)  (21111)   (321)
  (3111)    (111111)                                (111111)  (3111)
  (222)                                                       (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Columns k = 0 and k = n are A000041(n).
Column k = 1 and k = n-1 are A000041(n-1).
Column k = 2 appears to be 2*A027336(n).
The version for non-subset-sums is A046663, strict A365663.
Diagonal n = 2k is A108917, complement A366754.
Row sums are A304796, non-unique version A304792.
The non-unique version is A365543.
Cf. A002219, A122768, A275972, A299702, A299729, A301854, A364272, A364911, A365658, A365661, A367094.

Table[Length[Select[IntegerPartitions[n], Count[Total/@Union[Subsets[#]], k]==1&]], {n,0,10}, {k,0,n}]

A367094(n,1) = A108917(n).

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A304795 Number of positive special sums of the integer partition with Heinz number n.

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A299764 Number of special products of factorizations of n into factors > 1.

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A367108 Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.

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