### Brain Teasers

# Rnkfurers In Philos

Patty O'Furniture and her lopsided friend, Eileen, were down to their last dozen Rnkfurers. Rnkfurers are a little-known snack: a non-fat frankfurter whose natural flavorants include antivitriolic acids, sesquipedalian hydrocarbons, and golden bail powder. They're as nutritious as they are delicious, and the only advertising they need is word of mouth ... if you can get your friend to stop eating them long enough to tell you what they are.

Eileen was contemplating how many to buy; the 15 she'd bought hadn't lasted long, and only Patty's generosity had kept her from exorcising her craving by writing an acrostic sonnet about the treats during English class.

Now, after school, they were on their way to the local grocery store for their next order, strolling along in iambic pentameter. Patty plodded as Eileen's step provided the stresses. Gradually, they realized they could hear singing of a sort coming from up the street. The voices matched their pace -- one high and lilting in C-major, the other percussive and in the key of "off". As the two girls advanced, they tracked the sound to a large elm in Desire Creek's front yard. Patty spotted a red sneaker on one branch and grinned.

"I know that voice! C'mon down."

"Who is it?" Eileen prodded.

"Well, what do you call a guy making annoying noises in a tree?"

Eileen shrugged. "C'mon down, Jay," Patty called into the leaves.

"Why should I?"

"I'll give you each a Rnkfurer."

"Is that the new Sinatra CD?"

"No, it's the world's greatest snack."

"I'm hungry, too!" the other voice whined.

"Who's up there with you?" Patty asked.

"I'll bet I know!" Eileen called. "It's obvious!"

Patty's face lit up in realization. "Robin!" they laughed together.

A pair of hard landings and two bites later, Jay and Robin were negotiating to join the next bulk order of Rnkfurers. Patty explained how they'd always ordered such that one person got a number of Rnkfurers equal to the next person's percentage.

"But that doesn't work out very well with three people, and only with multiples of percentages for four," Eileen added.

"That's okay," Robin said. "Does it have to be percentages?"

"There's no law; just tradition. What do you have in mind?"

"We've just studied amicable numbers in algebra."

"What are those?"

"You know what a perfect number is?"

"Sure," Patty interjected, "a number equal to the sum of its divisors."

"Great," Jay continued. "Amicable numbers are a pair where each is equal to the sum of the other one's divisors."

Patty and Eileen digested that for a moment, then nodded.

"So, what do you have for four people?" Patty encouraged.

"I don't know. The smallest pair of amicable numbers is 220 and 284. Maybe we could order two sets of that?"

"I don't know; traditionally, it's been a dependent sequence. What's the next pair?"

"1184 and 1210."

"That's a lot, even for us. Besides, they come in cases of 25, 50, and 100, and we get a price break on the larger cases."

"Well, how about an aliquot sequence?"

"Okay. How does it taste?" Patty asked.

"Can you hum a few bars?" Eileen offered at the same time.

Robin laughed. "No, it's a series of numbers, each one being the sum of the previous number's divisors. For instance, start with 12. Its divisors add up to 16. 16's divisors add up to 15, then 9, and it's downhill from there."

Eileen and Patty understood. "So, you just don't have to get back to your starting point?"

"Right. Some of the sequences seem to go on for a long time, but a few stick in the same neighborhood for a few terms."

"Well, do you have any of those handy?"

"It depends; how many of these do we want to order? You two know what makes a week's supply, I hope?"

"Sure." Patty spoke as the veteran Rnkfurerphile. "At least 100 each, but probably not much more over the long haul."

Jay and Robin shared a glance. "Robin, is your homework still in your pack?"

"It sure is. One aliquot sequence coming up, the first four terms between 100 and 200, and the sum is divisible by 25."

Ten minutes later, the four friends were in the grocery store, ordering their Rnkfurers. What are the four quantities in the sequence?

Eileen was contemplating how many to buy; the 15 she'd bought hadn't lasted long, and only Patty's generosity had kept her from exorcising her craving by writing an acrostic sonnet about the treats during English class.

Now, after school, they were on their way to the local grocery store for their next order, strolling along in iambic pentameter. Patty plodded as Eileen's step provided the stresses. Gradually, they realized they could hear singing of a sort coming from up the street. The voices matched their pace -- one high and lilting in C-major, the other percussive and in the key of "off". As the two girls advanced, they tracked the sound to a large elm in Desire Creek's front yard. Patty spotted a red sneaker on one branch and grinned.

"I know that voice! C'mon down."

"Who is it?" Eileen prodded.

"Well, what do you call a guy making annoying noises in a tree?"

Eileen shrugged. "C'mon down, Jay," Patty called into the leaves.

"Why should I?"

"I'll give you each a Rnkfurer."

"Is that the new Sinatra CD?"

"No, it's the world's greatest snack."

"I'm hungry, too!" the other voice whined.

"Who's up there with you?" Patty asked.

"I'll bet I know!" Eileen called. "It's obvious!"

Patty's face lit up in realization. "Robin!" they laughed together.

A pair of hard landings and two bites later, Jay and Robin were negotiating to join the next bulk order of Rnkfurers. Patty explained how they'd always ordered such that one person got a number of Rnkfurers equal to the next person's percentage.

"But that doesn't work out very well with three people, and only with multiples of percentages for four," Eileen added.

"That's okay," Robin said. "Does it have to be percentages?"

"There's no law; just tradition. What do you have in mind?"

"We've just studied amicable numbers in algebra."

"What are those?"

"You know what a perfect number is?"

"Sure," Patty interjected, "a number equal to the sum of its divisors."

"Great," Jay continued. "Amicable numbers are a pair where each is equal to the sum of the other one's divisors."

Patty and Eileen digested that for a moment, then nodded.

"So, what do you have for four people?" Patty encouraged.

"I don't know. The smallest pair of amicable numbers is 220 and 284. Maybe we could order two sets of that?"

"I don't know; traditionally, it's been a dependent sequence. What's the next pair?"

"1184 and 1210."

"That's a lot, even for us. Besides, they come in cases of 25, 50, and 100, and we get a price break on the larger cases."

"Well, how about an aliquot sequence?"

"Okay. How does it taste?" Patty asked.

"Can you hum a few bars?" Eileen offered at the same time.

Robin laughed. "No, it's a series of numbers, each one being the sum of the previous number's divisors. For instance, start with 12. Its divisors add up to 16. 16's divisors add up to 15, then 9, and it's downhill from there."

Eileen and Patty understood. "So, you just don't have to get back to your starting point?"

"Right. Some of the sequences seem to go on for a long time, but a few stick in the same neighborhood for a few terms."

"Well, do you have any of those handy?"

"It depends; how many of these do we want to order? You two know what makes a week's supply, I hope?"

"Sure." Patty spoke as the veteran Rnkfurerphile. "At least 100 each, but probably not much more over the long haul."

Jay and Robin shared a glance. "Robin, is your homework still in your pack?"

"It sure is. One aliquot sequence coming up, the first four terms between 100 and 200, and the sum is divisible by 25."

Ten minutes later, the four friends were in the grocery store, ordering their Rnkfurers. What are the four quantities in the sequence?

### Hint

You can leave out numbers that are primes (sum of proper divisors is 1) or the product of two primes (sum of divisors will be less than 100). This will shorten your search somewhat.### Answer

108, 172, 136, and 134.Method:

This one really requires a brute-force approach. The sigma function (sum of a number's divisors) does not lend itself to 3rd-order analysis (3 numbers past the first one). Most of the remaining numbers will leave the 100-200 range on the first or second iteration. I used a small computer program. The only sequence that stays in bounds for 4 terms is the one given.

Side notes:

The name "Desire Creek" is a nod to "Desire Under the Elms" and the "Elm Creek Quilt" books.

There are 4-cycles of amicable numbers. However, the least quadruple is a little large, even for Patty's appetite: 1264460, 1547860, 1727636, 1305184.

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