A052273 Number of distinct 4th powers mod n.

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

Offset

1,2

Comments

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

S. Li, On the number of elements with maximal order in the multiplicative group modulo n, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1

Samer Seraj, Counting general power residues, Notes on Number Theory and Discrete Mathematics, 28:4 (2022), 730-743. Substituting k=4 into Theorem 1.1 gives a closed formula.

Samer Seraj, Resolution of Mathar's Conjectures on Counting Power Residues, #A62 INTEGERS 23 (2023).

Formula

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

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

From Samer Seraj, Nov 09 2022: (Start)

The above conjectures are correct, and a unified form is:

a(p^m) = alpha*((p^3 / p^beta)*((p^m - p^gamma)/(p^4 -1)) + ceiling((p^gamma)/(p^(beta+1)))) + 1, where p is any prime, m is any positive integer, alpha = (p-1)/gcd(4,p-1), beta = 3 if p = 2 or beta = 0 if p is odd, and gamma = 4 if 4|m or gamma = (m mod 4) otherwise. (End)

Maple

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

Mathematica

a[n_] := Table[PowerMod[i, 4, n], {i, 0, n-1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

Prog

(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
(PARI) \\ general formula for k-th powers, see Seraj link
h(p, e, k=4)=my(a=(p-1)/gcd(k, p-1), b=if(k%2+p%2, 0, valuation(k, p)+1)+p%2*valuation(k, p), g=(e-1)%k+1, G=p^g, B=p^(b+1), K=p^k, E=p^e); a*(K/B*(E-G)/(K-1)+ceil(G/B))+1
a(n, f=factor(n), k=4)=prod(i=1, #f~, h(f[i, 1], f[i, 2], k)) \\ Charles R Greathouse IV, Nov 09 2022

Crossrefs

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).

Sequence in context: A023155 A277847 A085311 * A369291 A074912 A274207

Adjacent sequences: A052270 A052271 A052272 * A052274 A052275 A052276

Keyword

nonn,mult

Author

N. J. A. Sloane, Feb 05 2000

Status

approved