A039706 -id:A039706 - cp's OEIS frontend

A007652 Final digit of prime(n).

2, 3, 5, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 3, 7, 1, 7, 9, 9, 1, 7, 3, 7, 3, 9, 1, 1, 3, 7, 9, 1, 3, 7, 9, 3, 9, 1, 1, 7, 3, 9, 1, 7, 1, 3, 3, 7, 1, 3, 7, 1, 7, 7, 9, 3, 9, 7, 3, 9, 3, 9, 7, 1, 9, 9, 1, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 7, 1, 9, 3, 9, 1, 3, 1, 7, 7, 3, 9, 1

Offset: 1

N. J. A. Sloane, Simon Plouffe

Primes modulo 10.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Alberto Fraile et al., Prime numbers and random walks in a square grid, Phys. Rev. E, 104(5) (2021), 054114.
Evelyn Lamb, "Peculiar Pattern Found in 'Random' Prime Numbers", Nature, March 14, 2016, republished by Scientific American.
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv preprint arXiv:1603.03720 [math.NT], 2016.
Index entries for sequences related to final digits of numbers.

Cf. A000040, A010879, A110923, A039701-A039706, A038194, A039709-A039715, A290450.

[Intseq(p)[1]: p in PrimesUpTo(600)]; // Bruno Berselli, Feb 14 2013

[p mod 10: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 10, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 10], {n, 120}] (* Ray Chandler, Oct 01 2005 *)
Mod[Prime[Range[100]], 10] (* Vincenzo Librandi, May 06 2014 *)

a(n)=prime(n)%10 \\ Charles R Greathouse IV, Oct 13 2016

primes(100)%10 \\ Charles R Greathouse IV, Oct 13 2016

a(n) = A010879(A000040(n)). - Michel Marcus, May 06 2014

Sum_k={1..n} a(k) ~ 5*n. - Amiram Eldar, Dec 11 2024

Extended by Ray Chandler, Oct 01 2005

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

Original entry on oeis.org

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

A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

Original entry on oeis.org

2, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 659, 673, 683

Offset: 1

N. J. A. Sloane

Sequence naturally partitions into two sequences: all primes p with ord_p(-2) odd (A163183, the primes dividing 2^j +1 for some odd j) and certain primes p with ord_p(-2) even (A163185). - Christopher J. Smyth, Jul 23 2009

Terms m in A047476 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002332, A002333.
Cf. A039706, A003628 (complement with respect to A000040).
Primes in A002479.
Cf. A051100 (see Mathar's comment).
Apart from leading term the same as A033200.

a033203 n = a033203_list !! (n-1)
a033203_list = filter ((== 1) . a010051) a047476_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

[p: p in PrimesUpTo(600) | p mod 8 in [1..3]]; // Vincenzo Librandi, Aug 11 2012

[p: p in PrimesUpTo(800) | NormEquation(2,p) eq true]; // Bruno Berselli, Jul 03 2016

QuadPrimes2[1, 0, 2, 10000] (* see A106856 *)
Select[Prime[Range[200]],MemberQ[{1,2,3},Mod[#,8]]&] (* Harvey P. Dale, Mar 16 2013 *)

is(n)=isprime(n) && issquare(Mod(-2,n)) \\ Charles R Greathouse IV, Nov 29 2016

a(n) = A002332(n) + 2*A002333(n)^2. - Zak Seidov, May 29 2014

A039702 a(n) = n-th prime modulo 4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Except for the first term, A100672(n) = (a(n)-1)/2 = parity of A005097. - Jeremy Gardiner, May 17 2008

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

Cf. A000040, A010873.
Cf. A005097, A100672.
Cf. A039701, A039703-A039706, A100672, A005097, A038194, A007652, A039709-A039715.

a039702 = (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 20 2012

[p mod 4: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 4, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 4], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 4] (* Vincenzo Librandi, May 06 2014 *)

a(n)=prime(n)%4 \\ Charles R Greathouse IV, Jun 13 2013

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

A003628 Primes congruent to {5, 7} mod 8.

Original entry on oeis.org

5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 263, 269, 271, 277, 293, 311, 317, 349, 359, 367, 373, 383, 389, 397, 421, 431, 439

Offset: 1

N. J. A. Sloane, Mira Bernstein

Inert rational odd primes in the field Q(sqrt(-2)).
Primes p such that p XOR 5 = p - 5. - Brad Clardy, Jul 22 2012
Terms m in A047566 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012
This sequence gives the primes p which satisfy norm(rho(p)) = - 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For p == 5 (mod 8) the norm is C(p, 0) (see a comment on 2*A230076) and for p == 7 (mod 8) the norm is -C(p, 0) (see a comment on A186302). For the primes with norm(rho(p)) = +1 see A033200. - Wolfdieter Lang, Oct 24 2013

H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to decomposition of primes in quadratic fields

Cf. A000040, A039706, A033203 (complement with respect to A000040).

a003628 n = a003628_list !! (n-1)
a003628_list = filter ((== 1) . a010051) a047566_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012

[ p: p in PrimesUpTo(600) | p mod 8 in {5, 7}]; // Vincenzo Librandi, Aug 22 2012

Select[Prime[Range[200]],MemberQ[{5,7},Mod[#,8]]&] (* Harvey P. Dale, Oct 24 2011 *)

{a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -2, m )==-1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 24 2023

A039703 a(n) = n-th prime modulo 5.

Original entry on oeis.org

2, 3, 0, 2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 3, 2, 1, 2, 4, 4, 1, 2, 3, 2, 3, 4, 1, 1, 3, 2, 4, 1, 3, 2, 4, 3, 4, 1, 1, 2, 3, 4, 1, 2, 1, 3, 3, 2, 1, 3, 2, 1, 2, 2, 4, 3, 4, 2, 3, 4, 3, 4, 2, 1, 4, 4, 1, 1, 3, 4, 3, 4, 2, 1, 3, 2, 4, 2, 1, 4, 3, 4, 1, 3, 1, 2, 2, 3, 4, 1

Offset: 1

Clark Kimberling

a(A049084(A045356(n-1))) = even; a(A049084(A045429(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008

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

Cf. A000040, A010874, A039701, A039702, A039704-A039706, A038194, A007652, A039709-A039715.

[p mod 5: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 5, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 5], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 5] (* Vincenzo Librandi, May 06 2014 *)

primes(105) % 5 \\ Zak Seidov, Apr 09 2013

Sum_k={1..n} a(k) ~ (5/2)*n. - Amiram Eldar, Dec 11 2024

A039715 Primes modulo 17.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 0, 2, 6, 12, 14, 3, 7, 9, 13, 2, 8, 10, 16, 3, 5, 11, 15, 4, 12, 16, 1, 5, 7, 11, 8, 12, 1, 3, 13, 15, 4, 10, 14, 3, 9, 11, 4, 6, 10, 12, 7, 2, 6, 8, 12, 1, 3, 13, 2, 8, 14, 16, 5, 9, 11, 4, 1, 5, 7, 11, 8, 14, 7, 9, 13, 2, 10, 16, 5

Offset: 1

Clark Kimberling

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

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

[p mod(17): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 17, n=1..100); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 17], {n, 100}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 17] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%17 \\ Charles R Greathouse IV, Apr 16 2012

[mod(p, 17) for p in primes(500)] # Bruno Berselli, May 05 2014

By the Prime Number Theorem in Arithmetic Progressions, all nonzero residue classes are equiprobable. In particular, Sum_{k=1..n} a(k) ~ 8.5n. - Charles R Greathouse IV, Apr 16 2012

A242119 Primes modulo 18.

Original entry on oeis.org

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

cp's OEIS Frontend

A007652 Final digit of prime(n).

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

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A033203 Primes p congruent to {1, 2, 3} (mod 8); or primes p of form x^2 + 2*y^2; or primes p such that x^2 = -2 has a solution mod p.

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A039702 a(n) = n-th prime modulo 4.

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A003628 Primes congruent to {5, 7} mod 8.

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A039703 a(n) = n-th prime modulo 5.

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