OFFSET
0,2
COMMENTS
For n > 0, a(n) is the unique number k in [1, 5^n] such that k^5 - 7 is divisible by 5^(n+1).
For k not divisible by 5, k is a fifth power in 5-adic field if and only if k == 1, 7, 18, 24 (mod 25). If k is a fifth power in 5-adic field, then k has exactly one fifth root.
LINKS
Wikipedia, p-adic number
EXAMPLE
For n = 5, we have 1047^5 - 7 = 5^6 * 80521782896, and that 1047 is the unique number k in [1, 5^5] such that k^5 - 7 is divisible by 5^6, so a(5) = 1047.
PROG
(PARI) a(n) = if(n, lift(sqrtn(7+O(5^(n+1)), 5)), 0)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 28 2019
STATUS
approved