12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061 |
- #!/usr/bin/ruby
- # Daniel "Trizen" Șuteu
- # Date: 11 February 2020
- # https://github.com/trizen
- # Fast recursive algorithm for generating all the k-powerful numbers <= n.
- # A positive integer n is considered k-powerful, if for every prime p that divides n, so does p^k.
- # Example:
- # 2-powerful = a^2 * b^3, for a,b >= 1
- # 3-powerful = a^3 * b^4 * c^5, for a,b,c >= 1
- # 4-powerful = a^4 * b^5 * c^6 * d^7, for a,b,c,d >= 1
- # OEIS:
- # https://oeis.org/A001694 -- 2-powerful numbers
- # https://oeis.org/A036966 -- 3-powerful numbers
- # https://oeis.org/A036967 -- 4-powerful numbers
- # https://oeis.org/A069492 -- 5-powerful numbers
- # https://oeis.org/A069493 -- 6-powerful numbers
- func k_powerful_numbers(n, k=2) {
- var powerful = []
- func (m,r) {
- if (r < k) {
- powerful << m
- return nil
- }
- for a in (1 .. iroot(idiv(n,m), r)) {
- if (r > k) {
- a.is_coprime(m) || next
- a.is_squarefree || next
- }
- __FUNC__(m * a**r, r-1)
- }
- }(1, 2*k - 1)
- powerful.sort
- }
- for k in (1..10) {
- say ("#{'%2d' % k}-powerful: ", k_powerful_numbers(5**k, k).join(', '))
- }
- __END__
- 1-powerful: 1, 2, 3, 4, 5
- 2-powerful: 1, 4, 8, 9, 16, 25
- 3-powerful: 1, 8, 16, 27, 32, 64, 81, 125
- 4-powerful: 1, 16, 32, 64, 81, 128, 243, 256, 512, 625
- 5-powerful: 1, 32, 64, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125
- 6-powerful: 1, 64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625
- 7-powerful: 1, 128, 256, 512, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 78125
- 8-powerful: 1, 256, 512, 1024, 2048, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 390625
- 9-powerful: 1, 512, 1024, 2048, 4096, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 1953125
- 10-powerful: 1, 1024, 2048, 4096, 8192, 16384, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 2097152, 4194304, 4782969, 8388608, 9765625
|