A039701 -id:A039701 - cp's OEIS frontend

A242119 Primes modulo 18.

2, 3, 5, 7, 11, 13, 17, 1, 5, 11, 13, 1, 5, 7, 11, 17, 5, 7, 13, 17, 1, 7, 11, 17, 7, 11, 13, 17, 1, 5, 1, 5, 11, 13, 5, 7, 13, 1, 5, 11, 17, 1, 11, 13, 17, 1, 13, 7, 11, 13, 17, 5, 7, 17, 5, 11, 17, 1, 7, 11, 13, 5, 1, 5, 7, 11, 7, 13, 5, 7, 11, 17, 7, 13, 1, 5

Offset: 1

Vincenzo Librandi, May 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. sequences of the type Primes mod k: A039701 (k=3), A039702 (k=4), A039703 (k=5), A039704 (k=6), A039705 (k=7), A039706 (k=8), A038194 (k=9), A007652 (k=10), A039709 (k=11), A039710 (k=12), A039711 (k=13), A039712 (k=14), A039713 (k=15), A039714 (k=16), A039715 (k=17), this sequence (k=18), A033633 (k=19), A242120(k=20), A242121 (k=21), A242122 (k=22), A229786 (k=23), A229787 (k=24), A242123 (k=25), A242124 (k=26), A242125 (k=27), A242126 (k=28), A242127 (k=29), A095959 (k=30), A110923 (k=100).

Magma
```
[p mod(18): p in PrimesUpTo(500)];
```
Mathematica
```
Mod[Prime[Range[100]], 18]
```

[mod(p, 18) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_{i=1..n} a(i) ~ 9n. The derivation is the same as in the formula in A039715. - Jerzy R Borysowicz, Apr 27 2022

A039704 a(n) = n-th prime modulo 6.

Original entry on oeis.org

2, 3, 5, 1, 5, 1, 5, 1, 5, 5, 1, 1, 5, 1, 5, 5, 5, 1, 1, 5, 1, 1, 5, 5, 1, 5, 1, 5, 1, 5, 1, 5, 5, 1, 5, 1, 1, 1, 5, 5, 5, 1, 5, 1, 5, 1, 1, 1, 5, 1, 5, 5, 1, 5, 5, 5, 5, 1, 1, 5, 1, 5, 1, 5, 1, 5, 1, 1, 5, 1, 5, 5, 1, 1, 1, 5, 5, 1, 5, 1, 5, 1, 5, 1, 1, 5, 5, 1, 5, 1, 5, 5, 1, 5, 1, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1

Offset: 1

Clark Kimberling

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

Cf. A000040, A039701-A039703, A039705, A039706, A038194, A007652, A039709-A039715.

[p mod 6: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 6, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 6], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 6] (* Vincenzo Librandi, May 06 2014 *)

primes(105)%6 \\ Zak Seidov, Mar 25 2012

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

A039705 a(n) = n-th prime modulo 7.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(A049084(A045370(n-1))) is even; a(A049084(A045415(n-1))) is odd. - Reinhard Zumkeller, Feb 25 2008

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

Cf. A039701-A039704, A039706, A038194, A007652, A039709-A039715.

[p mod(7): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 7, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 7], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 7] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%7 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ (7/2)*n. - Amiram Eldar, Dec 11 2024

A134323 a(n) = Legendre(-3, prime(n)).

Original entry on oeis.org

-1, 0, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1

Offset: 1

Reinhard Zumkeller, Oct 21 2007

Value of lowest trit of prime(n) in balanced ternary representation (A059095) (original definition).

For p = prime(n) != 3, a(n) = +1 if p is of the form 3*k + 1, and -1 if the p is of the form 3*k - 1. - Joerg Arndt, Sep 16 2014

			For n=20, prime(20) = 71, and we verify that -3 is not a quadratic residue modulo 71, hence a(20) = -1. Also, we see that the balanced ternary representation row A059095(71) = {1, 0, -1, 0, -1} which ends in -1.
For n=21, prime(21) = 73, and we see that x^2 = -3 mod 73 has solutions like x = 17, 56, hence a(21) = 1. Also, the balanced ternary representation row A059095(73) = {1, 0 -1, 0, 1} which ends in 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A039701, A049084, A112632 (partial sums), A059095 (balanced ternary)
Cf. A091177 (indices of -1's), A091178 (indices of +1's), A003627, A002476.
Other moduli: A070750, A257834.
Cf. A102283.

a134323 n = (1 - 0 ^ m) * (-1) ^ (m + 1) where m = a000040 n `mod` 3
-- Reinhard Zumkeller, Sep 16 2014

A134323[n_] := (r = Mod[Prime[n], 6]; If[r == 1, 1, -1]); A134323[1] = -1; A134323[2] = 0; Table[A134323[n], {n, 1, 102}] (* Jean-François Alcover, Nov 07 2011, after Bill McEachen *)
JacobiSymbol[-3, Prime[Range[100]]] (* Alonso del Arte, Aug 02 2017 *)

apply(p->kronecker(-3,p), primes(100)) \\ Charles R Greathouse IV, Aug 14 2017

-1 if the n-th prime is 2 or == 5 mod 6, +1 if the n-th prime is == 1 mod 6, and 0 if it is 3.
a(n) = (1 - 0^A039701(n)) * (-1)^(A039701(n)+1).
a(n) != 0 for n != 2;
a(A049084(A003627(n))) = -1; a(A049084(A002476(n))) = +1.
a(n) = A102283(prime(n)). - Ridouane Oudra, Jan 09 2025

Wording of definition changed by N. J. A. Sloane, Jun 21 2015

Name simplified by Alonso del Arte, Aug 02 2017

A103566 Sum of the primes > 5 modulo 3.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 13, 15, 16, 18, 20, 22, 23, 24, 26, 27, 28, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 46, 47, 49, 50, 51, 52, 54, 56, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 72, 74, 75, 77, 79, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100, 101, 103, 105, 106

Offset: 1

Roger L. Bagula, Mar 23 2005

Cf. A039701, A103567-A103572, A120326.

a = Table[Sum[Mod[Prime[i + 3], 3], {i, 1, n}], {n, 1, 200}]
Accumulate[Mod[Prime[Range[4,80]],3]] (* Harvey P. Dale, Aug 04 2013 *)

a(n+1)-a(n) = A039701(n+4) (first differences).

a(n) = A120326(n+3) - 4. - Jason Yuen, Sep 01 2024

A039710 a(n) = n-th prime modulo 12.

Original entry on oeis.org

2, 3, 5, 7, 11, 1, 5, 7, 11, 5, 7, 1, 5, 7, 11, 5, 11, 1, 7, 11, 1, 7, 11, 5, 1, 5, 7, 11, 1, 5, 7, 11, 5, 7, 5, 7, 1, 7, 11, 5, 11, 1, 11, 1, 5, 7, 7, 7, 11, 1, 5, 11, 1, 11, 5, 11, 5, 7, 1, 5, 7, 5, 7, 11, 1, 5, 7, 1, 11, 1, 5, 11, 7, 1, 7, 11, 5, 1, 5, 1, 11

Offset: 1

Clark Kimberling

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

Cf. A039701-A039706, A038194, A007652, A039709, A039711-A039715.

[p mod(12): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 12, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 12], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 12] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%11 \\ Charles R Greathouse IV, Dec 12 2024

[mod(p, 12) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 6*n. - Amiram Eldar, Dec 11 2024

A039711 a(n) = n-th prime modulo 13.

Original entry on oeis.org

2, 3, 5, 7, 11, 0, 4, 6, 10, 3, 5, 11, 2, 4, 8, 1, 7, 9, 2, 6, 8, 1, 5, 11, 6, 10, 12, 3, 5, 9, 10, 1, 7, 9, 6, 8, 1, 7, 11, 4, 10, 12, 9, 11, 2, 4, 3, 2, 6, 8, 12, 5, 7, 4, 10, 3, 9, 11, 4, 8, 10, 7, 8, 12, 1, 5, 6, 12, 9, 11, 2, 8, 3, 9, 2, 6, 12, 7, 11, 6

Offset: 1

Clark Kimberling

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

Cf. A039701-A039706, A038194, A007652, A039709, A039710, A039712-A039715.

[p mod(13): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 13, n=1..100); # Nathaniel Johnston, Jun 29 2011

Mod[ Prime@ Range@ 80, 13] (* Robert G. Wilson v, Mar 13 2011 *)

primes(1000)%13 \\ Charles R Greathouse IV, Mar 13 2011

[mod(p, 13) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ (13/2)*n. - Amiram Eldar, Dec 11 2024

A039712 a(n) = n-th prime modulo 14.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 3, 5, 9, 1, 3, 9, 13, 1, 5, 11, 3, 5, 11, 1, 3, 9, 13, 5, 13, 3, 5, 9, 11, 1, 1, 5, 11, 13, 9, 11, 3, 9, 13, 5, 11, 13, 9, 11, 1, 3, 1, 13, 3, 5, 9, 1, 3, 13, 5, 11, 3, 5, 11, 1, 3, 13, 13, 3, 5, 9, 9, 1, 11, 13, 3, 9, 3, 9, 1, 5, 11, 5, 9

Offset: 1

Clark Kimberling

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

Cf. A039701-A039706, A038194, A007652, A039709-A039711, A039713-A039715.

[p mod(14): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 14, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 14], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 14] (* Vincenzo Librandi, May 06 2014 *)

[mod(p, 14) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 7*n. - Amiram Eldar, Dec 12 2024

A039713 a(n) = n-th prime modulo 15.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 2, 4, 8, 14, 1, 7, 11, 13, 2, 8, 14, 1, 7, 11, 13, 4, 8, 14, 7, 11, 13, 2, 4, 8, 7, 11, 2, 4, 14, 1, 7, 13, 2, 8, 14, 1, 11, 13, 2, 4, 1, 13, 2, 4, 8, 14, 1, 11, 2, 8, 14, 1, 7, 11, 13, 8, 7, 11, 13, 2, 1, 7, 2, 4, 8, 14, 7, 13, 4, 8, 14, 7

Offset: 1

Clark Kimberling

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

Cf. A039701-A039706, A038194, A007652, A039709-A039712, A039714, A039715.

[p mod(15): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 15, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 15], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 15] (* Vincenzo Librandi, May 06 2014 *)

[mod(p, 15) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ (15/2)*n. - Amiram Eldar, Dec 12 2024

A039714 a(n) = n-th prime modulo 16.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 1, 3, 7, 13, 15, 5, 9, 11, 15, 5, 11, 13, 3, 7, 9, 15, 3, 9, 1, 5, 7, 11, 13, 1, 15, 3, 9, 11, 5, 7, 13, 3, 7, 13, 3, 5, 15, 1, 5, 7, 3, 15, 3, 5, 9, 15, 1, 11, 1, 7, 13, 15, 5, 9, 11, 5, 3, 7, 9, 13, 11, 1, 11, 13, 1, 7, 15, 5, 11, 15, 5

Offset: 1

Clark Kimberling

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

Cf. A039701-A039706, A038194, A007652, A039709-A039713, A039715.

[p mod(16): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 16, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 16], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 16] (* Vincenzo Librandi, May 06 2014 *)

[mod(p, 16) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_k={1..n} a(k) ~ 8*n. - Amiram Eldar, Dec 12 2024

cp's OEIS Frontend

A242119 Primes modulo 18.

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A039704 a(n) = n-th prime modulo 6.

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A039705 a(n) = n-th prime modulo 7.

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A134323 a(n) = Legendre(-3, prime(n)).

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A103566 Sum of the primes > 5 modulo 3.

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A039710 a(n) = n-th prime modulo 12.

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Magma

Maple

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Sage

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A039711 a(n) = n-th prime modulo 13.

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Magma