number.wiki

Term

Happy Number

Iterating the sum of squares of digits eventually reaches 1 (1, 7, 10, 13, 19, 23, 28, 31, 32, 44, …).

1,718 numbers tagged.

A happy number is one for which iterating the sum of squares of its decimal digits eventually reaches 1. Numbers that don't are unhappy — they fall into the cycle \(4 \to 16 \to 37 \to 58 \to 89 \to 145 \to 42 \to 20 \to 4\) and repeat forever.

Example: \(19 \to 1^2 + 9^2 = 82 \to 8^2 + 2^2 = 68 \to 6^2 + 8^2 = 100 \to 1\). So 19 is happy.

Happy primes are primes that are also happy. The first are 7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193. Like other digit-based properties, happiness is base-dependent.

74,381 74,413 74,489 74,507 74,531 74,587 74,611 74,699 74,831 74,857 75,041 75,169 75,223 75,277 75,401 75,407 75,431 75,553 75,619 75,731 75,767 75,781 75,787 76,001 76,003 76,091 76,159 76,303 76,519 76,603 76,757 76,913 76,949 76,961 77,093 77,101 77,153 77,201 77,239 77,261 77,269 77,351 77,513 77,527 77,587 77,621 78,041 78,059 78,157 78,283 78,341 78,401 78,439 78,509 78,517 78,571 78,577 78,721 78,823 79,111 79,229 79,273 79,349 79,399 79,493 79,561 79,601 79,613 79,627 79,631 79,843 79,939 79,943 80,141 80,147 80,233 80,251 80,369 80,407 80,471 80,603 80,909 80,929 80,963 81,041 81,047 81,223 81,401 81,421 81,611 81,727 81,839 81,899 82,051 82,141 82,153 82,231 82,237 82,261 82,339 82,351 82,387 82,393 82,463 82,531 82,549 82,561 82,651 82,723 82,787 82,837 82,939 83,003 83,023 83,203 83,221 83,227 83,299 83,417 83,471 83,497 83,609 83,891 84,011 84,017 84,121 84,131 84,137 84,211 84,223 84,263 84,317 84,401 84,449 84,701 84,713 84,731 84,793 84,947 84,979 85,021 85,201 85,213 85,429 85,453 85,621 85,717 85,781 85,817 85,933 86,111 86,243 86,369 86,423 86,441 86,693 86,851 87,041 87,223 87,509 87,517 87,547 87,721 87,751 87,943 88,001 88,237 88,327 88,651 88,729 88,801 88,853 88,919 88,969 89,009 89,057 89,069 89,189 89,209 89,293 89,329 89,381 89,533 89,603 89,689 89,819 89,891 89,923 90,001 90,007 90,023 90,031 90,073 90,089 90,203 90,263 90,289 90,313 90,617 90,703