Photo: Alex Green

What numbers make people happy?

The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, ... (OEIS A007770). These are also the numbers whose 2-recurring digital invariant sequences have period 1.

mathworld.wolfram.com - Happy Number -- from Wolfram MathWorld

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Let the sum of the squares of the digits of a positive integer be represented by . In a similar way, let the sum of the squares of the digits of be represented by , and so on. Iterating this sum-of-squared-digits map always eventually reaches one of the 10 numbers 0, 1, 4, 16, 20, 37, 42, 58, 89, or 145 (OEIS A039943; Porges 1945). If for some , then the original integer is said to be happy. For example, starting with 7 gives the sequence 7, 49, 97, 130, 10, 1, so 7 is a happy number. The first few happy numbers are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, ... (OEIS A007770). These are also the numbers whose 2-recurring digital invariant sequences have period 1. The numbers of iterations required for these to reach 1 are 0, 5, 1, 2, 4, 3, 3, 2, 3, 4, 4, 2, 5, ... (OEIS A090425). The numbers of happy numbers less than or equal to 1, , , ... are given by 1, 3, 20, 143, 1442, 14377, 143071, ... (OEIS A068571). The first few consecutive happy numbers have , 129, 192, 262, 301, 319, 367, 391, ... (OEIS A035502). Similarly, the first few happy triplets start with 1880, 4780, 4870, 7480, 7839, ... (OEIS A072494). The first few happy primes are 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, ... (OEIS A035497). Once it is known whether a number is happy (or not), then any number in the sequence , , , ... will also be happy (or not). A number that is not happy is called unhappy. Unhappy numbers have eventually periodic sequences of which never reach 1. Any permutation of the digits of an unhappy or happy number must also be unhappy or happy. This follows from the fact that addition is commutative.

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What is the hardest part of being rich?

The hardest part of becoming a billionaire isn't keeping tabs on your accounts, finding friends and employees you can trust, or juggling the demands on your deep pockets. It's getting started.

businessinsider.com - This Is The Hardest Part Of Getting Rich, According To A Billionaire

Billionaire Ted Turner is responsible for major television networks including CNN. Chip Somodevilla/Getty The hardest part of becoming a billionaire isn't keeping tabs on your accounts, finding friends and employees you can trust, or juggling the demands on your deep pockets.

It's getting started.

At least, that was the hardest part for media magnate and billionaire Ted Turner. In "Billionaires: Reflections On The Upper Crust," Darrell M. West recalls Turner's 1995 address to Brown University, where West was teaching political science.

West describes a key point in Turner's presentation:

His most difficult challenge, he said, had been making the first million dollars; after that, everything was easy. Money begets money, he bluntly observed, thereby making it possible to gain even greater wealth through social and political connections. West uses this comment as a jumping off point for a discussion of how the wealthiest citizens in the US wield undue influence on its government. "This, of course, is the crux of the controversy about the role of billionaires in society," he writes. Not every recollection from Turner's visit to Brown was so heavy, however: He also talked about raising Buffalo in Montana, and joked that his favorite part of ranch life was the ability to "take a whiz" off his front porch.

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