A301970 -id:A301970 - cp's OEIS frontend

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

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

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).

cp's OEIS Frontend

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

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