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A301979 Number of subset-sums minus number of subset-products of the integer partition with Heinz number n. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 1 02:00 EDT 2020. Contains 333153 sequences. (Running on oeis4.)