A192453 -id:A192453 - cp's OEIS frontend

A047232 Smallest k > n where n is a 4th power modulo k.

1, 2, 7, 6, 6, 10, 10, 9, 14, 11, 15, 14, 22, 17, 22, 21, 17, 19, 21, 27, 22, 25, 26, 29, 29, 27, 37, 33, 31, 46, 34, 33, 46, 34, 37, 38, 38, 41, 43, 42, 41, 43, 47, 49, 49, 55, 49, 53, 52, 53, 62, 59, 54, 58, 57, 57, 61, 59, 61, 62, 66, 65, 67, 74, 66, 67, 75, 77, 76, 73, 74

Offset: 0

Henry Bottomley, Jun 08 2000

a(n) = n+1 for n+1 in A192453. - Robert Israel, May 24 2019

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A192453.

N:= 100: # for a(0)..a(N)
V:= Array(0..N):
count:= 1: V[0]:= 1:
for k from 2 while count < N+1 do
  R:= select(t -> t <= N and V[t] = 0,{seq(x^4 mod k,x=1..k/2)});
  V[convert(R,list)]:= k;
  count:= count + nops(R);
od:
convert(V,list); # Robert Israel, May 24 2019

a(n) = {my(k=n+1); while (!ispower(Mod(n,k), 4), k++); k;} \\ Michel Marcus, May 25 2019

A192452 Numbers n such that -1 is not a 4th power mod n.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

José María Grau Ribas, Jul 01 2011

nonn

Complement of A192453.

Table[If[Reduce[x^4==-1,Modulus->n]===False,n],{n,2,300}]//Union

A227784 Least number of fourth powers which add to -1 mod n.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 2, 7, 2, 4, 2, 3, 2, 2, 4, 15, 1, 2, 2, 4, 2, 2, 2, 7, 4, 2, 2, 3, 3, 4, 2, 15, 2, 1, 4, 3, 2, 2, 2, 7, 1, 2, 2, 3, 4, 2, 2, 15, 2, 4, 2, 3, 2, 2, 4, 7, 2, 3, 2, 4, 2, 2, 2, 15, 4, 2, 2, 3, 2, 4, 2, 7, 1, 2, 4, 3, 2, 2, 2, 15, 2, 1, 2, 3, 4, 2, 3, 7, 1, 4, 2

Offset: 1

Charles R Greathouse IV, Aug 01 2013

nonn

Parnami, Agrawal, & Rajwade proved (1981, Theorem 1) that, for a prime p > 29, a(p) = 1 if p = 1 mod 8 and otherwise a(p) = 2.

Conjecture: a(n) = 15 if n = 9 mod 16 and a(n) = 7 if n = 8 mod 16, otherwise a(n) <= 4. (The associated lower bounds are obvious.)

J. C. Parnami, M. K. Agrawal, and A. R. Rajwade, On the 4-power Stufe of a field, Rendiconti del Circolo Matematico di Palermo (2) 30:2 (1981), pp. 245-254.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. C. Parnami, M. K. Agrawal, and A. R. Rajwade, On the 4-power Stufe of p-adic completions of algebraic number fields, Rendiconti del Seminario Matematico Università e Politecnico di Torino 44:1 (1986), pp. 141-153.

Cf. A192453, A227781.

a(n)=if(n==1,return(0)); if(n>29 && isprime(n), return(if(n%8>1, 2, 1))); my(N,cur,new,k=1);for(i=1,n\2,cur=N=bitor(1<<(i^4%n),N));while(!bittest(cur,n-1),new=0;for(i=1,n\2,t=cur<<(i^4%n);t=bitor(bitand(t,(1<>n);new=bitor(new,t));k++;cur=new);k

a(n) <= A002377(n-1) <= 19.

a(n) = 1 if and only if n > 1 is in A192453.

A319838 Numbers k such that A319100(k) is a power of 6.

Original entry on oeis.org

1, 2, 7, 9, 13, 14, 18, 19, 26, 27, 31, 37, 38, 43, 49, 54, 61, 62, 63, 67, 73, 74, 79, 81, 86, 91, 97, 98, 103, 109, 117, 122, 126, 127, 133, 134, 139, 146, 151, 157, 158, 162, 163, 169, 171, 181, 182, 189, 193, 194, 199, 206, 211, 217, 218, 223, 229, 234, 241

Offset: 1

Jianing Song, Sep 28 2018

nonn

Numbers k such that the number of solutions to x^6 == 1 (mod k) is a power of 6.
Also numbers k such that (Z/kZ)* has the same 2-rank and 3-rank, where (Z/kZ)* is the multiplicative group of integers modulo k, and the p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) with p being a prime number.
k is a term in this sequence iff v(2, k) = 0 or 1, v(3, k) = 0 or >= 2 and k is not divisible by any prime p == 5 (mod 6). Here v(p, k) is the p-adic valuation of k.
Sequence contains all primes p == 1 (mod 6) and their powers and all powers of 3 except for 3 itself.
Decompose the multiplicative group of integers modulo k as a product of cyclic groups C_{s_1} x C_{s_2} x ... x C_{s_m}, where s_i divides s_j for i < j, then k is a term iff s_1 is divisible by 6. For k = 1 or 2, (Z/kZ)* is the trivial group, s_1 does not exist so 1 and 2 are also terms.
If gcd(k_1, k_2) = 1 and both k_1 and k_2 are in this sequence, so is k_1*k_2. For example, 7 and 9 are both here so 7*9 = 63 is also here. Indeed, the number of solutions to x^6 == 1 (mod 7), x^6 == 1 (mod 9) and x^6 == 1 (mod 36) are 6, 6 and 36, respectively.
This is an analog of A008784, since k is a term there iff s_1 (defined as above) is divisible by 4 instead of 6. But on the other hand, if k is in A008784, so are all its divisors, while this is not true for this sequence. However, if k is here and k is not divisible by 9, then all its divisors are also here.
This is a also an analog of A192453 (s_1 divisible by 8).

			91 = 7*13 is a term since the number of solutions to x^6 == 1 (mod 91) is 36 = 6^2.
1197 = 7*9*19 is a term since the number of solutions to x^6 == 1 (mod 1197) is 216 = 6^3.

Jianing Song, Table of n, a(n) for n = 1..15917 (all terms below 100000)

Cf. A008784, A192453, A319100.

isA319838(n) = if(abs(n)==1||abs(n)==2, 1, my(i=znstar(n)[2]); !(i[#i]%6)==1, 1)

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A047232 Smallest k > n where n is a 4th power modulo k.

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A192452 Numbers n such that -1 is not a 4th power mod n.

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A227784 Least number of fourth powers which add to -1 mod n.

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A319838 Numbers k such that A319100(k) is a power of 6.

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