FunWithNumbers (version 1.2)

sptable: Calculate the number of unique values in the cross-table of sums and products for the input set of numbers

Description

This function tests the proposition that the sum of all unique values in the cross-table of sums and products for a set of N input values is "close" to N^2 .

Usage

sptable(x)

Value

.

uniqsum

vector of the unique values of the outer sum outer(x,x,'+')

.

uniqprod

vector of the unique values of the outer product outer(x,x)

.

spratio

The ratio uniqsum/uniquprod

exponentOfN

The (numeric) solution to N^(exponentOfN) = uniqsum+uniqprod. If Erdos is right, this will always be "close" to 2.

Arguments

x

A vector of integer values.

Author

Carl Witthoft,

References

This conjecture is discussed in https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/

Examples

(sptable(1:10))
# $uniqsum
# [1] 19
# $uniqprod
# [1] 42
# $spratio
# [1] 0.452381
# $exponentOfN
# [1] 1.78533
set.seed(42)
sptable(sample(1:100,20,rep=FALSE))
# $uniqsum
# [1] 123
# $uniqprod
# [1] 202
# $spratio
# [1] 0.6089109
# $exponentOfN
# [1] 1.930688