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A322157 The successive approximations up to 5^n for 5-adic integer 7^(1/5). 5
0, 2, 22, 47, 422, 1047, 13547, 29172, 341672, 732297, 732297, 30029172, 127685422, 860107297, 4522216672, 10625732297, 41143310422, 498906982297, 1261846435422, 5076543701047, 62297002685422, 348399297607297, 1778910772216672, 8931468145263547, 20852397100341672 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..24.

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

For the digits of this number see A322169.

For fifth roots in 7-adic field, see A309450, A309451, A309452, A309453, A309454.

Sequence in context: A080142 A306969 A200946 * A156454 A168669 A271883

Adjacent sequences:  A322154 A322155 A322156 * A322158 A322159 A322160

KEYWORD

nonn

AUTHOR

Jianing Song, Aug 28 2019

STATUS

approved

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Last modified January 22 10:13 EST 2022. Contains 350481 sequences. (Running on oeis4.)