Happy numbers, Recreational Mathematics and Dr. Who

Last night we watched Dr. Who.  "In the 2007 Doctor Who episode "42", a sequence of happy primes (313, 331, 367, 379) is used as a code for unlocking a sealed door on a spaceship about to collide with a star. When the Doctor learns that nobody on the spaceship besides himself has heard of happy numbers, he asks, "Don't they teach recreational mathematics anymore?""   (Happy number - Wikipedia, the free encyclopedia)

I learned about happy prime numbers.   Dr. Who said, "Any number that reduces to one when you take the sum of the square of its digits and continue iterating it until it yields 1 is a happy number, any number that doesn't, isn't. A happy prime is both happy and prime."

What?!

The Wikipedia article sited above clarified with some examples:

For example, 19 is happy, as the associated sequence is:
1² + 9² = 82
8² + 2² = 68
6² + 8² = 100
1² + 0² + 0² = 1.
The 143 happy numbers up to 1,000 are:
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000 (sequence A007770 in OEIS).

Very cool!