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

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

1,4

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

Formula

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

Example

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.

Mathematica

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}]

Prog

(PARI)
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

Crossrefs

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

Sequence in context: A090330 A332447 A132747 * A183063 A318979 A172441

Adjacent sequences: A301976 A301977 A301978 * A301980 A301981 A301982

Keyword

sign

Author

Gus Wiseman, Mar 30 2018

Status

approved