A066601 -id:A066601 - cp's OEIS frontend

A096385 a(n) = smallest prime p with p^n mod n = 1.

3, 7, 3, 11, 5, 29, 3, 7, 11, 23, 5, 53, 13, 31, 3, 103, 5, 191, 3, 37, 23, 47, 5, 11, 53, 7, 13, 59, 11, 311, 3, 67, 67, 71, 5, 149, 37, 61, 3, 83, 5, 173, 23, 31, 47, 283, 5, 29, 11, 103, 5, 107, 5, 31, 13, 7, 59, 709, 7, 367, 61, 37, 3, 131, 23, 269, 13, 139, 29, 569

Offset: 2

Reinhard Zumkeller, Aug 05 2004

nonn

			n=5: 2^5=32=5*6+2, 3^5=243=5*48+3, 5^5 mod 5 = 0, 7^5=16807=5*3361+2, 11^5=161051=5*32210+1: a(5)=11.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A015910, A066601, A066603, A066438, A066441.

With[{prs=Prime[Range[200]]},Table[SelectFirst[prs,PowerMod[#,n,n]==1&],{n,2,80}]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Aug 31 2015 *)

a(n) = my(p=2); while (Mod(p,n)^n !=1, p=nextprime(p+1)); p; \\ Michel Marcus, Feb 07 2021

A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 3, 4, 1, 0, 1, 0, 1, 4, 3, 2, 1, 0, 0, 1, 0, 0, 4, 3, 0, 1, 0, 1, 2, 1, 1, 1, 4, 1, 8, 1, 0, 0, 0, 0, 2, 0, 5, 0, 0, 4, 1, 0, 1, 1, 1, 3, 1, 6, 1, 1, 9, 2, 1, 0, 0, 2, 0, 4, 4, 0, 0, 8, 6, 3, 4, 1, 0, 1, 0, 1, 0, 3, 1, 1, 0, 5, 4, 9, 2, 1

Offset: 1

Leroy Quet, Feb 14 2006

Alternate description: triangular array a(n, k) = n^k (mod k) read by rows (n > 1, 0 < k < n). This is equivalent because a(n, k) = a(n-k, k). - David Wasserman, Jan 25 2007

			2^6 = 64 and 64 (mod 6) is 4. So a(2,6) = 4.

Cf. A051127, A051128, A094263. Transpose of A060154. First 13 rows are A057427(n-1), A015910, A066601, A066602, A066603, A066604, A066438, A066439, A066440, A056969, A066441, A066442, A116609. First 12 columns are A000004, A000035, A010872, A000035, A010874, A070431, A010876, A000035, A021559(n+1), A008959, A010880, A070435.

a[n_, k_] := Mod[n^k, k]; Table[a[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

More terms from David Wasserman, Jan 25 2007

A374911 a(n) = a(2^n mod n) + a(3^n mod n), with a(0) = 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 7, 7, 7, 3, 4, 7, 7, 7, 7, 7, 10, 3, 7, 11, 7, 5, 10, 7, 7, 7, 18, 7, 8, 21, 7, 7, 7, 3, 11, 7, 18, 25, 7, 7, 11, 5, 7, 17, 7, 10, 18, 7, 7, 14, 14, 21, 11, 10, 7, 29, 14, 7, 11, 7, 7, 13, 7, 7, 11, 3, 17, 7, 7, 10, 11, 21, 7, 7, 7, 7, 21, 10, 32, 11, 7, 5, 6, 7, 7, 14, 10, 7, 11, 19

Offset: 0

Bryle Morga, Jul 23 2024

Conjectured to contain all positive integers. Here are the indexes where each of the first few positive integers appear:
0
1
2, 4, 8, 16, 32, ... (2^k, k > 0)
3, 9, ...
20, 40, 80, 272, 320, 328, ...
81, 66469, 144937, ...
5, 6, 7, 10, 11, 12, 13,... (all primes appear except 2 and 3)
27, 301, 729, 1099, 2107, 2187, 85085, 1594323, ...
Most solutions to a(n) = 5 seem to be divisible by 5 and all of them seem to be even. Why?
Are 3 and 9 the only solutions to a(n) = 4?

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

Cf. A000079, A015910, A066601.

a[0]=1; a[n_]:=a[PowerMod[2,n,n]]+a[PowerMod[3,n,n]]; Array[a,89,0] (* Stefano Spezia, Jul 23 2024 *)

a(n) = if (n==0, 1, a(lift(Mod(2,n)^n)) + a(lift(Mod(3,n)^n))); \\ Michel Marcus, Jul 25 2024

def a(n):
  return 1 if n == 0 else a(pow(2, n, n)) + a(pow(3, n, n))

a(p) = 7 for primes p except 2 and 3.

a(2^n) = 3 for n > 0.

cp's OEIS Frontend

A096385 a(n) = smallest prime p with p^n mod n = 1.

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A114448 Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).

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A374911 a(n) = a(2^n mod n) + a(3^n mod n), with a(0) = 1.

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