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A052273 Number of distinct 4th powers mod n. 14


%S 1,2,2,2,2,4,4,2,4,4,6,4,4,8,4,2,5,8,10,4,8,12,12,4,6,8,10,8,8,8,16,4,

%T 12,10,8,8,10,20,8,4,11,16,22,12,8,24,24,4,22,12,10,8,14,20,12,8,20,

%U 16,30,8,16,32,16,6,8,24,34,10,24,16,36,8,19,20,12,20

%N Number of distinct 4th powers mod n.

%C This sequence is multiplicative [Li]. - Leon P Smith, Apr 16 2005

%H T. D. Noe, <a href="/A052273/b052273.txt">Table of n, a(n) for n=1..1000</a>

%H S. Li, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav86i2p113bwm">On the number of elements with maximal order in the multiplicative group modulo n</a>, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1

%F Conjecture: a(2^e) = 1+floor[2^e/(2^4-1)] if e ==0 (mod 4). a(2^e) = 2+floor[2^e/(2^4-1)] if e == {1,2,3} mod 4. - _R. J. Mathar_, Oct 22 2017

%F Conjecture: a(p^e) = 1+floor[ (p-1)*p^(e+3)/{gcd(p-1,4)*(p^4-1)}] for odd primes p. - _R. J. Mathar_, Oct 22 2017

%p A052273 := proc(n,k) local i; nops({seq(i^k mod n,i=0..n-1)}); end; # number of k-th powers mod n

%o (PARI) a(n)=my(f=factor(n)); prod(i=1,#f[,1],my(k=f[i,1]^f[i,2]); #vecsort(vector(k,i,i^4%k),,8)) \\ _Charles R Greathouse IV_, May 26 2013

%Y Cf. A000224 (squares), A046530 (cubic residues), A052274 (5th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_, Feb 05 2000

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Last modified April 19 18:37 EDT 2019. Contains 322288 sequences. (Running on oeis4.)