A217659 -id:A217659 - cp's OEIS frontend

A039701 a(n) = n-th prime modulo 3.

2, 0, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1

Offset: 1

Clark Kimberling

If n > 2 and prime(n) is a Mersenne prime then a(n) = 1. Proof: prime(n) = 2^p - 1 for some odd prime p, so prime(n) = 2*4^((p-1)/2) - 1 == 2 - 1 = 1 (mod 3). - Santi Spadaro, May 03 2002; corrected and simplified by Dean Hickerson, Apr 20 2003
Except for n = 2, a(n) is the smallest number k > 0 such that 3 divides prime(n)^k - 1. - T. D. Noe, Apr 17 2003
a(n) <> 0 for n <> 2; a(A049084(A003627(n))) = 2; a(A049084(A002476(n))) = 1; A134323(n) = (1 - 0^a(n)) * (-1)^(a(n)+1). - Reinhard Zumkeller, Oct 21 2007
Probability of finding 1 (or 2) in this sequence is 1/2. This follows from the Prime Number Theorem in arithmetic progressions. Examples: There are 4995 1's in terms (10^9 +1) to (10^9+10^4); there are 10^9/2-1926 1's in the first 10^9 terms. - Jerzy R Borysowicz, Mar 06 2022

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A091178 (indices of 1's), A091177 (indices of 2's).
Cf. A120326 (partial sums).
Cf. A185934, A217659.
Cf. A010872.
Other moduli: A039702-A039706, A038194, A007652, A039709-A039715.

a039701 = (`mod` 3) . a000040
a039701_list = map (`mod` 3) a000040_list
-- Reinhard Zumkeller, Nov 16 2012

[p mod(3): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 3, n=1..105); # Nathaniel Johnston, Jun 29 2011

Mathematica
```
Table[Mod[Prime[n], 3], {n, 100}]
```

primes(100)%3 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ (3/2)*n. - Amiram Eldar, Dec 11 2024

A185934 Lesser of two consecutive primes which both equal 1 (mod 3).

Original entry on oeis.org

31, 61, 73, 151, 157, 199, 211, 271, 331, 367, 373, 433, 523, 541, 571, 601, 607, 619, 661, 727, 733, 751, 991, 997, 1033, 1063, 1069, 1117, 1123, 1201, 1231, 1237, 1291, 1321, 1381, 1453, 1459, 1531, 1543, 1621, 1657, 1669, 1741, 1747, 1753, 1759, 1777, 1789, 1861, 1987, 2011, 2131, 2161, 2179, 2281, 2287, 2341, 2371

Offset: 1

M. F. Hasler, Feb 06 2011

nonn

Or, primes of the form 6k+1 such that the next prime is again of the form 6k'+1.

			The smallest prime of the form 6k+1 such that the next larger prime differs by a multiple of 3 (and thus a multiple of 6), is a(1) = 31, the following prime being 31+6 = 37.
Note that the next larger prime may also differ by 12 (as is the case for 199,211,619,661,997,1201,1237,1459,1531,1789,3049,...), or by 18 (as it is the case for 523,1069,1381,1759,2161,2503,3889,...), etc.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002476, A151800, A217659, A219244.

a185934 n = a185934_list !! (n-1)
a185934_list = map (a000040 . (+ 1)) $
   elemIndices 1 $ zipWith (*) a039701_list $ tail a039701_list
-- Reinhard Zumkeller, Nov 16 2012

With[{p = Prime[Range[350]]}, ind = Position[Mod[p, 3], 1] // Flatten; p[[ind[[Position[Differences[ind], 1] // Flatten]]]]] (* Amiram Eldar, Apr 12 2025 *)

forprime( p=1,1e4, (o+0-o=p)%3==0 & o%3==1 & print1( precprime(p-1)","))

a(n) = A217659(n) - 6*A219244(n); A217659(n) = A151800(a(n)). - Reinhard Zumkeller, Nov 16 2012

A219244 Differences of two consecutive primes which both equal 1 modulo 3, divided by 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 16 2012

nonn

a(n) = (A217659(n) - A185934(n)) / 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A185934, A217659.

a219244 n = a219244_list !! (n-1)
a219244_list = map (`div`  6) $ zipWith (-) a217659_list a185934_list

(Last[#]-First[#])/6&/@Select[Partition[Prime[Range[800]],2,1], Mod[First[#],3] == Mod[Last[#],3]==1&] (* Harvey P. Dale, Feb 26 2013 *)

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A039701 a(n) = n-th prime modulo 3.

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A185934 Lesser of two consecutive primes which both equal 1 (mod 3).

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A219244 Differences of two consecutive primes which both equal 1 modulo 3, divided by 6.

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