A066520 -id:A066520 - cp's OEIS frontend

A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n.

0, 1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of primes <= n that are quadratic nonresidues modulo 11 minus the number of primes <= n that are quadratic residues modulo 11.
It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 138 ones are negative.
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			Below 200, there are 20 primes congruent to 1, 3, 4, 5, 9 modulo 11 and 23 primes congruent to 2, 6, 7, 8, 10 modulo 11, so a(200) = 23 - 20 = 3.

Wikipedia, Chebyshev's bias

Cf. A112632.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: this sequence (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, isprime(i)*kronecker(-11, i))

a(n) = -Sum_{primes p<=n} Legendre(p,11) = -Sum_{primes p<=n} Kronecker(-11,p) = -Sum_{primes p<=n} A011582(p).

A321862 a(n) = A321857(prime(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 3, 4, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 7, 6, 5, 4, 5, 4, 5, 4

Offset: 1

Jianing Song, Nov 20 2018

sign

The first 10000 terms are positive, but conjecturally infinitely many terms should be negative.
The first negative term occurs at a(102091236) = -1. - Jianing Song, Nov 08 2019
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			prime(25) = 97, Pi(5,1)(97) = Pi(5,4)(97) = 5, Pi(5,2)(97) = Pi(5,3)(97) = 7, so a(25) = 7 + 7 - 5 - 5 = 4.

Wikipedia, Chebyshev's bias

Cf. A080891.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), this sequence (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, kronecker(5, prime(i)))

a(n) = -Sum_{i=1..n} Legendre(prime(i),5) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A080891(prime(i)).

Edited by Peter Munn, Nov 19 2023

A321864 a(n) = A321859(prime(n)).

Original entry on oeis.org

-1, 0, 1, 1, 0, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 5, 6, 5, 4, 5, 4, 5, 4, 5, 6, 5, 4, 5, 6, 5, 4

Offset: 1

Jianing Song, Nov 20 2018

sign

Among the first 10000 terms there are only 13 negative ones, with the earliest one (besides a(1)) being a(5006) = -1.

Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			prime(25) = 97. Among the primes <= 97, there are 10 ones congruent to 1, 2, 4 modulo 7 and 14 ones congruent to 3, 5, 6 modulo 7, so a(25) = 14 - 10 = 4.

Wikipedia, Chebyshev's bias

Cf. A175629.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), this sequence (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, kronecker(-7, prime(i)))

a(n) = -Sum_{primes p<=n} Legendre(prime(i),7) = -Sum_{primes p<=n} Kronecker(-7,prime(i)) = -Sum_{i=1..n} A175629(prime(i)).

A321865 a(n) = A321860(prime(n)).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

Among the first 10000 terms there are only 32 negative ones.

Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			prime(46) = 199. Among the primes <= 199, there are 20 ones congruent to 1, 3, 4, 5, 9 modulo 11 and 23 ones congruent to 2, 6, 7, 8, 10 modulo 11, so a(46) = 23 - 20 = 3.

Wikipedia, Chebyshev's bias

Cf. A011582.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: this sequence (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, kronecker(-11, prime(i)))

a(n) = -Sum_{primes p<=n} Legendre(prime(i),11) = -Sum_{primes p<=n} Kronecker(-11,prime(i)) = -Sum_{i=1..n} A011582(prime(i)).

A051024 Values of n for which pi_{4,3}(p_n) - pi_{4,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Original entry on oeis.org

2946, 50378, 50380, 50382, 50392, 50414, 50418, 50420, 50422, 50424, 50426, 50428, 50430, 50436, 50438, 50446, 50448, 50450, 50822, 50832, 50834, 50842, 50844, 50852, 50854, 50856, 50858, 50862, 50864, 50866, 50872, 50892, 50902

Offset: 1

Eric W. Weisstein

nonn

This is a companion sequence to A051025.
Starting from a(27556) = 316064952540 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 418933 terms (see a-file) with a(418933) = 330797040308 as its last term. - Sergei D. Shchebetov, Oct 06 2017
We also discovered the 9th sign-changing zone, which starts from 2083576475506, ends with 2083615410040, and has 13370 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017
We also discovered the 10th sign-changing zone, which starts from 21576098946648, ends with 22056324317296, and has 481194 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 28 2018

Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55.
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp.54-76.
M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp.1565-1575.
A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, Pages 173-197.
Sergei D. Shchebetov, First 418933 terms (zipped file)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547.

Cf. A156749 (Sequence showing Chebyshev bias in prime races (mod 4)). - Daniel Forgues, Mar 26 2009

For[i=2; d=0, True, i++, d+=Mod[Prime[i], 4]-2; If[d==-1, Print[i]]]
(* Second program: *)
Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* Michael De Vlieger, Dec 30 2017 *)

from sympy import nextprime; a, p = 0, 2
for n in range(2, 50917):
    p=nextprime(p); a += p%4-2
    if a == -1: print(n, end = ', ') # Ya-Ping Lu, Jan 18 2025

Edited by Dean Hickerson, Mar 05 2002

A321858 a(n) = Pi(12,5)(n) + Pi(12,7)(n) - Pi(12,1)(n) - Pi(12,11)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of odd primes <= n that have 3 as a quadratic nonresidue minus the number of primes <= n that have 3 as a quadratic residue.
The first 10000 terms are nonnegative. a(p) = 0 for primes p = 2, 3, 13, 433, 443, 457, 479, 491, 503, 3541, ... The earliest negative term is a(61463) = -1. Conjecturally infinitely many terms should be negative.
In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Nov 19 2023]
Here, although 11 is not a quadratic residue modulo 12, for most n we have Pi(12,7)(n) + Pi(12,11)(n) > Pi(12,1)(n) - Pi(12,5)(n), Pi(12,5)(n) + Pi(12,11)(n) > Pi(12,1)(n) + Pi(12,7)(n) and Pi(12,5)(n) + Pi(12,7)(n) > Pi(12,1)(n) + Pi(12,11)(n).

			Pi(12,1)(100) = 5, Pi(12,5)(100) = Pi(12,7)(100) = Pi(12,11)(100) = 6, so a(100) = 6 + 6 - 5 - 6 = 1.

Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias

Cf. A007350, A110161.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), this sequence (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, isprime(i)*kronecker(12, i))

a(n) = -Sum_{primes p<=n} Kronecker(12,p) = -Sum_{primes p<=n} A110161(p).

A321861 a(n) = A071838(prime(n)).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 5, 6, 5, 4, 3, 4, 3, 2, 3, 2, 3, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 7, 6, 5, 4, 5, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 4, 5, 6, 5, 6, 7, 6, 5, 6, 7, 6, 5, 4, 5, 4

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative.
The first negative term occurs at a(732722) = -1. - Jianing Song, Nov 08 2019
In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Nov 19 2023].
Here, although 7 is not a quadratic residue modulo 8, for most n we have Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) - Pi(8,3)(n), Pi(8,3)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,5)(n) and Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,7)(n).

			prime(25) = 97, Pi(8,1)(97) = 5, Pi(8,3)(97) = 7, Pi(8,5)(97) = Pi(8,7)(97) = 6, so a(25) = 7 + 6 - 5 - 6 = 2.

Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias

Cf. A007350, A091337.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), this sequence (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, kronecker(2, prime(i)))

a(n) = -Sum_{i=1..n} Kronecker(prime(i),2) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A091337(prime(i)).

Edited by Peter Munn, Nov 19 2023

A321863 a(n) = A321858(prime(n)).

Original entry on oeis.org

0, 0, 1, 2, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 5, 6, 5, 6, 5, 6, 5, 4, 3, 2, 3, 4, 5, 6, 5, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 3, 2, 1, 0, 1, 0, 1

Offset: 1

Jianing Song, Nov 20 2018

sign

Among the first 10000 terms there are only 291 negative ones, with the earliest one being a(6181) = -1. See the comments about "Chebyshev's bias" in A321858.

			prime(25) = 97, Pi(12,1)(97) = 5, Pi(12,5)(97) = Pi(12,7)(97) = Pi(12,11)(97) = 6, so a(25) = 6 + 6 - 5 - 6 = 1.

Wikipedia, Chebyshev's bias

Cf. A110161.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), this sequence (d=12).

a(n) = -sum(i=1, n, kronecker(12, prime(i)))

a(n) = -Sum_{primes p<=n} Kronecker(12,prime(i)) = -Sum_{i=1..n} A110161(prime(i)).

A096628 Integers k for which {prime(1), prime(2), ..., prime(k)} (mod 4) is biased towards 1.

Original entry on oeis.org

2946, 50378, 50380, 50382, 50383, 50384, 50385, 50386, 50387, 50388, 50389, 50390, 50391, 50392, 50414, 50415, 50416, 50417, 50418, 50419, 50420, 50421, 50422, 50424, 50426, 50428, 50430, 50436, 50438, 50446, 50447, 50448, 50450

Offset: 1

Eric W. Weisstein, Jul 01 2004

nonn

Indices of negative terms in A038698. - Jianing Song, Feb 20 2019

Robert Israel, Table of n, a(n) for n = 1..7804
Eric Weisstein's World of Mathematics, Chebyshev Bias

Cf. A007350, A038691, A038698, A051024, A051025, A066520, A199547.

R:= NULL: count:= 0: v:= 0: p:= 2:
for i from 2 while count < 100 do
  p:= nextprime(p);
  if p mod 4 = 3 then v:= v+1
  else v:= v-1
  fi;
  if v < 0 then R:= R,i; count:= count+1 fi
od:
R; # Robert Israel, Apr 17 2023

A199547 Primes p for which pi_{4,3}(p) < pi_{4,1}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Original entry on oeis.org

26861, 616841, 616849, 616877, 616897, 616909, 616933, 616943, 616951, 616961, 616991, 616997, 616999, 617011, 617269, 617273, 617293, 617311, 617327, 617333, 617339, 617341, 617359, 617369, 617401, 617429, 617453, 617521, 617537, 617689, 617693, 617699, 617717

Offset: 1

Arkadiusz Wesolowski, Dec 09 2011

nonn

Another version of A007350.
J. E. Littlewood (1914) proved that this sequence is infinite.
a(1) = 26861 was found in 1957 by John Leech.
Prime indices of negative terms in A066520. - Jianing Song, Feb 20 2019

Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 22.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 26861

Cf. A007350, A038691, A038698, A051024, A051025, A066520, A096628.

lst = {}; For[n = 2; t = 0, n < 50451, n++, t += Mod[p = Prime[n], 4] - 2; If[t < 0, AppendTo[lst, p]]]; lst

from sympy import nextprime; a, p = 0, 2
while p < 617717:
    p=nextprime(p); a += p%4-2
    if a < 0: print(p, end = ', ') # Ya-Ping Lu, Jan 18 2025

a(n) = prime(A096628(n)). - Jianing Song, Feb 20 2019

cp's OEIS Frontend

A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A321862 a(n) = A321857(prime(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A321864 a(n) = A321859(prime(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A321865 a(n) = A321860(prime(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A051024 Values of n for which pi_{4,3}(p_n) - pi_{4,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A321858 a(n) = Pi(12,5)(n) + Pi(12,7)(n) - Pi(12,1)(n) - Pi(12,11)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A321861 a(n) = A071838(prime(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions