A301854 -id:A301854 - cp's OEIS frontend

A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			42 is the Heinz number of (4,2,1) which is strict and knapsack, so is in the sequence. 45 is the Heinz number of (3,2,2) which is knapsack but not strict, so is not in the sequence. 30 is the Heinz number of (3,2,1) which is strict but not knapsack, so is not in the sequence.
Sequence of strict knapsack partitions begins: (), (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (8), (42), (51), (9), (61).

Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301900.

wt[n_]:=If[n===1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
Select[Range[100],SquareFreeQ[#]&&UnsameQ@@wt/@Divisors[#]&]

Intersection of A299702 and A005117.

A301957 Number of distinct subset-products of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 29 2018

nonn

A subset-product of an integer partition y is a product of some submultiset of y. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Number of distinct values obtained when A003963 is applied to all divisors of n. - Antti Karttunen, Sep 05 2018

			The distinct subset-products of (4,2,1,1) are 1, 2, 4, and 8, so a(84) = 4.
The distinct subset-products of (6,3,2) are 1, 2, 3, 6, 12, 18, and 36, so a(195) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000712, A001055, A001227, A002865, A003963, A108917, A162247, A276024, A292886, A301854, A301855, A301856, A301970, A301979, A304793.

Table[If[n===1,1,Length[Union[Times@@@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

up_to = 65537;
A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
v003963 = vector(up_to,n,A003963(n));
A301957(n) = { my(m=Map(),s,k=0,c); fordiv(n,d,if(!mapisdefined(m,s = v003963[d],&c), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 05 2018

More terms from Antti Karttunen, Sep 05 2018

A301855 Number of divisors d|n such that no other divisor of n has the same Heinz weight A056239(d).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

			The a(24) = 4 special divisors are 1, 2, 12, 24.

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

Cf. A000712, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301830, A301854, A301855, A301856.

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

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A301855(n) = if(1==n,n,my(m=Map(),w,s); fordiv(n,d,w = A056239(d); if(!mapisdefined(m, w, &s), mapput(m,w,Set([d])), mapput(m,w,setunion(Set([d]),s)))); sumdiv(n,d,(1==length(mapget(m,A056239(d)))))); \\ Antti Karttunen, Jul 01 2018

More terms from Antti Karttunen, Jul 01 2018

A301900 Heinz numbers of strict non-knapsack partitions. Squarefree numbers such that more than one divisor has the same Heinz weight A056239(d).

Original entry on oeis.org

30, 70, 154, 165, 210, 273, 286, 330, 390, 442, 462, 510, 546, 561, 570, 595, 646, 690, 714, 741, 770, 858, 870, 874, 910, 930, 1045, 1110, 1122, 1155, 1173, 1190, 1230, 1254, 1290, 1326, 1330, 1334, 1365, 1410, 1430, 1482, 1495, 1590, 1610, 1653, 1770

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of strict non-knapsack partitions begins: (321), (431), (541), (532), (4321), (642), (651), (5321), (6321), (761), (5421), (7321), (6421), (752), (8321), (743), (871), (9321), (7421), (862), (5431), (6521).

Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301899.

wt[n_]:=If[n===1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
Select[Range[1000],SquareFreeQ[#]&&!UnsameQ@@wt/@Divisors[#]&]

Complement of A005117 in A299702.

A301935 Number of positive subset-sum trees whose composite a positive subset-sum of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 10, 2, 3, 1, 21, 1, 3, 3, 58, 1, 21, 1, 21, 3, 3, 1, 164, 2, 3, 10, 21, 1, 34, 1, 373, 3, 3, 3, 218, 1, 3, 3, 161, 1, 7, 1, 5, 5, 3, 1, 1320, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 4, 2558, 3, 7, 1, 5, 3, 6, 1, 7

Offset: 1

Gus Wiseman, Mar 28 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A positive subset-sum tree with root x is either the symbol x itself, or is obtained by first choosing a positive subset-sum x <= (y_1,...,y_k) with k > 1 and then choosing a positive subset-sum tree with root y_i for each i = 1...k. The composite of a positive subset-sum tree is the positive subset-sum x <= g where x is the root sum and g is the multiset of leaves. We write positive subset-sum trees in the form rootsum(branch,...,branch). For example, 4(1(1,3),2,2(1,1)) is a positive subset-sum tree with composite 4(1,1,1,2,3) and weight 8.

Gus Wiseman, The a(12) = 21 positive subset-sum trees.

Cf. A000108, A000712, A108917, A122768, A262671, A262673, A275972, A276024, A284640, A299701, A301854, A301855, A301856, A301934.

A301979 Number of subset-sums minus number of subset-products of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 30 2018

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A subset-sum (or subset-product) of a multiset y is any number equal to the sum (or product) of some submultiset of y.
First negative entry is a(165) = -1.
This sequence is unbounded above and below.

			The distinct subset-sums of (4,2,1,1) are 0, 1, 2, 3, 4, 5, 6, 7, 8, while the distinct subset-products are 1, 2, 4, 8, so a(84) = 9 - 4 = 5.
The distinct subset-sums of (5,3,2) are 0, 2, 3, 5, 7, 8, 10, while the distinct subset-products are 1, 2, 3, 5, 6, 10, 15, 30, so a(165) = 7 - 8 = -1.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000712, A003963, A056239, A108917, A276024, A284640, A296150, A299701, A301854, A301855, A301856, A301957, A301970.

Table[With[{ptn=If[n===1,{},Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Length[Union[Plus@@@Subsets[ptn]]]-Length[Union[Times@@@Subsets[ptn]]]],{n,100}]

A003963(n) = {n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n)};
A301957(n) = {my(ds = divisors(n)); for(i=1,#ds,ds[i] = A003963(ds[i])); #Set(ds)};
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A299701(n) = {my(ds = divisors(n)); for(i=1,#ds,ds[i] = A056239(ds[i])); #Set(ds)};
A301979(n) = (A299701(n) - A301957(n)); \\ Antti Karttunen, Oct 07 2018

a(n) = A299701(n) - A301957(n).

A301934 Number of positive subset-sum trees of weight n.

Original entry on oeis.org

1, 3, 14, 85, 586, 4331, 33545, 268521, 2204249

Offset: 1

Gus Wiseman, Mar 28 2018

A positive subset-sum tree with root x is either the symbol x itself, or is obtained by first choosing a positive subset-sum x <= (y_1,...,y_k) with k > 1 and then choosing a positive subset-sum tree with root y_i for each i = 1...k. The weight is the sum of the leaves. We write positive subset-sum trees in the form rootsum(branch,...,branch). For example, 4(1(1,3),2,2(1,1)) is a positive subset-sum tree with composite 4(1,1,1,2,3) and weight 8.

			The a(3) = 14 positive subset-sum trees:
3           3(1,2)       3(1,1,1)     3(1,2(1,1))
2(1,2)      2(1,1,1)     2(1,1(1,1))  2(1(1,1),1)  2(1,2(1,1))
1(1,2)      1(1,1,1)     1(1,1(1,1))  1(1(1,1),1)  1(1,2(1,1))

Cf. A000108, A000712, A108917, A122768, A262671, A262673, A275972, A276024, A284640, A299701, A301854, A301855, A301856, A301935.

A304796 Number of special sums of integer partitions of n.

Original entry on oeis.org

1, 2, 5, 10, 18, 32, 51, 82, 122, 188, 262, 392, 529, 750, 997, 1404, 1784, 2452, 3123, 4164, 5239, 6916, 8499, 11112, 13693, 17482, 21257, 27162, 32581, 41114, 49606, 61418, 73474, 91086, 107780, 132874, 157359, 191026, 225159, 274110, 320691, 386722, 453875

Offset: 0

Gus Wiseman, May 18 2018

nonn

A special sum of an integer partition y is a number n >= 0 such that exactly one submultiset of y sums to n.

			The a(4) = 18 special positive subset-sums:
0<=(4), 4<=(4),
0<=(22), 2<=(22), 4<=(22),
0<=(31), 1<=(31), 3<=(31), 4<=(31),
0<=(211), 1<=(211), 3<=(211), 4<=(211),
0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854.

uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

a(n) = A301854(n) + A000041(n).

More terms from Alois P. Heinz, May 18 2018

a(36)-a(42) from Chai Wah Wu, Sep 26 2023

A321144 Irregular triangle where T(n,k) is the number of divisors of n whose prime indices sum to k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Gus Wiseman, Oct 28 2018

The rows are all palindromes.

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.

			Triangle begins:
  1
  1  1
  1  0  1
  1  1  1
  1  0  0  1
  1  1  1  1
  1  0  0  0  1
  1  1  1  1
  1  0  1  0  1
  1  1  0  1  1
  1  0  0  0  0  1
  1  1  2  1  1
  1  0  0  0  0  0  1
  1  1  0  0  1  1
  1  0  1  1  0  1
  1  1  1  1  1
  1  0  0  0  0  0  0  1
  1  1  1  1  1  1
  1  0  0  0  0  0  0  0  1
  1  1  1  1  1  1
  1  0  1  0  1  0  1
  1  1  0  0  0  1  1
  1  0  0  0  0  0  0  0  0  1
  1  1  2  2  1  1
  1  0  0  1  0  0  1
  1  1  0  0  0  0  1  1
  1  0  1  0  1  0  1
  1  1  1  0  1  1  1
  1  0  0  0  0  0  0  0  0  0  1
  1  1  1  2  1  1  1

Row lengths are A056239. Number of nonzero entries in row n is A299701(n).

Cf. A000005, A112798, A276024, A301854, A301899.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
Table[Count[Total/@primeMS/@Divisors[n],k],{n,20},{k,0,Total[primeMS[n]]}]

A301970 Heinz numbers of integer partitions with more subset-products than subset-sums.

Original entry on oeis.org

165, 273, 325, 351, 495, 525, 561, 595, 675, 741, 765, 819, 825, 931, 1045, 1053, 1155, 1173, 1425, 1485, 1495, 1575, 1625, 1653, 1683, 1771, 1785, 1815, 1911, 2025, 2139, 2145, 2223, 2275, 2277, 2295, 2310, 2415, 2457, 2465, 2475, 2625, 2639, 2695, 2805, 2945

Offset: 1

Gus Wiseman, Mar 29 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A subset-sum (or subset-product) of a multiset y is any number equal to the sum (or product) of some submultiset of y.

Numbers n such that A301957(n) > A299701(n).

			Sequence of partitions begins: (532), (642), (633), (6222), (5322), (4332), (752), (743), (33222), (862), (7322), (6422), (5332), (844), (853), (62222), (5432), (972), (8332), (53222), (963), (43322), (6333).

Cf. A000712, A003963, A056239, A108917, A276024, A284640, A292886, A296150, A299701, A299702, A299729, A301854, A301855, A301856, A301957, A301979.

Select[Range[1000],With[{ptn=If[#===1,{},Join@@Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Length[Union[Times@@@Subsets[ptn]]]>Length[Union[Plus@@@Subsets[ptn]]]]&]

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A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

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A301957 Number of distinct subset-products of the integer partition with Heinz number n.

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A301855 Number of divisors d|n such that no other divisor of n has the same Heinz weight A056239(d).

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A301900 Heinz numbers of strict non-knapsack partitions. Squarefree numbers such that more than one divisor has the same Heinz weight A056239(d).

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A301935 Number of positive subset-sum trees whose composite a positive subset-sum of the integer partition with Heinz number n.

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A301979 Number of subset-sums minus number of subset-products of the integer partition with Heinz number n.

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A301934 Number of positive subset-sum trees of weight n.

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A304796 Number of special sums of integer partitions of n.

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