A098044 -id:A098044 - cp's OEIS frontend

A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

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

Offset: 1

Roger Hui, Dec 22 2005

Cumulative sums of A134323, negated. The first negative term is a(23338590792) = -1 for the prime 608981813029. See page 4 of the paper by Granville and Martin. - T. D. Noe, Jan 23 2008 [Corrected by Jianing Song, Nov 24 2018]

See the comment about "Chebyshev's bias" in A321856. - Jianing Song, Nov 24 2018

			a(1) = 1 because 2 == -1 (mod 3).
a(2) = 1 because 3 == 0 (mod 3) and does not change the counting.
a(3) = 2 because 5 == -1 (mod 3).
a(4) = 1 because 7 == 1 (mod 3).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), pp. 1-33.
Wikipedia, Chebyshev's bias

Cf. A007352, A098044, A102283, A134323.
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), this sequence (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a112632 n = a112632_list !! (n-1)
a112632_list = scanl1 (+) $ map negate a134323_list
-- Reinhard Zumkeller, Sep 16 2014

a[n_] := a[n] = a[n-1] + If[Mod[Prime[n], 6] == 1, -1, 1]; a[1] = a[2] = 1; Table[a[n], {n, 1, 100}]  (* Jean-François Alcover, Jul 24 2012 *)
Accumulate[Which[IntegerQ[(#+1)/3],1,IntegerQ[(#-1)/3],-1,True,0]& /@ Prime[ Range[100]]] (* Harvey P. Dale, Jun 06 2013 *)

a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ Jianing Song, Nov 24 2018

a(n) = -Sum_{primes p<=n} Legendre(prime(i),3) = -Sum_{primes p<=n} Kronecker(-3,prime(i)) = -Sum_{i=1..n} A102283(prime(i)). - Jianing Song, Nov 24 2018

A007352 Where the prime race 3k-1 vs. 3k+1 changes leader.

Original entry on oeis.org

2, 608981813029, 608981813507, 608981813683, 608981813819, 608981814127, 608981814143, 608981818999, 608981820977, 608981826877, 608981826977, 608981827873, 608981828201, 608981836363, 608981836493, 608981836681, 608981836973, 608981836993, 608981837063

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, Apr 28 1994

nonn

Terms a(2n+1) form a subsequence of A098044.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence, although the terms are incorrect - see A185703).

Donovan Johnson, Table of n, a(n) for n = 1..39215
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2}(x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
G. Martin, Asymmetries in the Shanks-Renyi Prime number race, arXiv:math/0010086 [math.NT], 2000.
Pieter Moree, Chebyshev's bias for composite numbers with restricted prime divisors, Mathematics of computation 73.245 (2004): 425-449. See page 425.
M. Rubinstein and P. Sarnak, Chebyshev's Bias, Exper. Math. 3 (4) (1994) 209.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A098044, A297406, A297407, A297408, A297410, A297411.

Terms from a(3) onwards corrected by Max Alekseyev, Feb 10 2011

A096629 Values of n for which {p_3, p_4, ..., p_n} (mod 3) contains equal numbers of 1's and 2's.

Original entry on oeis.org

4, 6, 8, 12, 14, 22, 38, 48, 50, 23338590786, 23338590788, 23338590790, 23338590806, 23338590808, 23338590820, 23338590822, 23338590824, 23338590826, 23338590828, 23338590830, 23338590834, 23338590840, 23338590858, 23338590860, 23338590868, 23338590870

Offset: 1

Eric W. Weisstein, Jul 04 2004

nonn

Donovan Johnson, Table of n, a(n) for n = 1..85508
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2}(x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
Eric Weisstein's World of Mathematics, Chebyshev Bias

Cf. A000720, A096630, A096449, A098044, A096452.

a(n) = A000720(A098044(n+1)).

Edited by Max Alekseyev, Sep 13 2009

Terms a(10) onward from Max Alekseyev, Feb 10 2011

A096449 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is equal to the number of such primes congruent to 2 mod 3.

Original entry on oeis.org

5, 11, 17, 23, 41, 47, 83, 167, 227, 233, 608981812919, 608981812961, 608981813017, 608981813569, 608981813677, 608981813833, 608981813851, 608981813927, 608981813939, 608981813963, 608981814043, 608981814149, 608981814251, 608981814827

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

First term prime(3) = 5 is placed on 0th row.
If prime(n-1) = +1 mod 3 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 mod 3 is on k-th row then we put prime(n) on (k+1)-st row.
This process makes an array of prime numbers:
5, 11, 17, 23, 41, 47, 83, ... (this sequence)
7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101, .. (A096452).
31, 59, 67, 73, 97, ... (A096453)
61, ...

Cf. A096448-A096455.

lst = {5}; p = 0; q = 0; r = 5; While[r < 10^9, If[ Mod[r, 3] == 2, p++, q++ ]; r = NextPrime@r; If[p == q, AppendTo[lst, r]; Print@r]]; lst (* Robert G. Wilson v, Sep 20 2009 *)

For n>1, a(n) = prime(A096629(n-1)+1) = A000040(A096629(n-1)+1). - Max Alekseyev, Sep 19 2009

a(n) = A151800(A098044(n)) = A007918(A098044(n)+1).

More terms and better definition from Joshua Zucker, May 21 2006

Terms a(11) onward from Max Alekseyev, Feb 10 2011

A096452 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is one less than the number of such primes congruent to 2 mod 3.

Original entry on oeis.org

7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101, 107, 113, 131, 163, 173, 223, 229, 239, 251, 383, 443, 1811, 1871, 1877, 1889, 608981812613, 608981812667, 608981812891, 608981812951, 608981812993, 608981813929, 608981813941, 608981814019, 608981814173

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
G. D. Martin, Biases in the Shanks-Renyi prime number race, FoCM'02 Conference, Minneapolis, 5-14 Aug 2002.

Cf. A096447-A096455, A096629, A098044.

lst = {}; p = q = 0; r = 5; While[r < 10^5, If[ Mod[r, 3] == 2, p++, q++ ]; r = NextPrime@ r; If[p == q + 1, AppendTo[lst, r]; Print@ r]]; lst (* Robert G. Wilson v, Sep 20 2009 *)

More terms and better definition from Joshua Zucker, May 21 2006

Terms a(27) onward from Max Alekseyev, Feb 10 2011

A297006 Primes p for which pi_{3,2}(p) - pi_{3,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Original entry on oeis.org

608981813029, 608981813137, 608981813261, 608981813273, 608981813311, 608981813357, 608981813459, 608981813683, 608981813717, 608981813777, 608981813789, 608981814127, 608981818999, 608981819021, 608981819273, 608981819359, 608981819419, 608981820869, 608981820899, 608981820913, 608981826877, 608981827873, 608981827891, 608981828023, 608981828029, 608981828111, 608981828129, 608981836363, 608981836391, 608981836481

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 23 2017

nonn

This sequence is a companion sequence to A297005. Starting from a(20591)=6148171711663 the sequence includes the second sign-changing zone predicted by C. Bays et al. in 2001. The sequence with the first two sign-changing zones up to 10^13 contains 84323 terms with a(84323)=6156051951677 as its last term (see b-file). In addition, a(1) = A007352(2) as well as a(20591) = A007352(9630).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..84323
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2} (x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576
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, pp. 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007352, A096629, A096630, A096449, A098044.

A297005 Values of n for which pi_{3,2}(p_n) - pi_{3,1}(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

23338590792, 23338590794, 23338590796, 23338590798, 23338590800, 23338590802, 23338590804, 23338590810, 23338590814, 23338590816, 23338590818, 23338590832, 23338591016, 23338591018, 23338591028, 23338591030, 23338591032, 23338591084, 23338591086, 23338591088, 23338591302, 23338591340, 23338591342, 23338591344, 23338591346, 23338591348, 23338591350, 23338591656, 23338591658, 23338591662

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 23 2017

nonn

This sequence is a companion sequence to A297006. Starting from a(20591)=216415270060 the sequence includes the second sign-changing zone predicted by C. Bays et al. in 2001. The sequence with the first two sign-changing zones up to 10^13 contains 84323 terms with a(84323)=216682882512 as its last term (see b-file). In addition, a(1) = A096630(1).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..84323
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2} (x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
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, pp. 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007352, A096629, A096630, A096449, A098044.

A174695 Partial sums of A173950.

Original entry on oeis.org

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

Offset: 1

Giovanni Teofilatto, Nov 30 2010

sign

Since this sequence equals A112632(n)-1, and A007352 gives the primes at which the sign of A112632 changes, we have a change of sign in the present sequence not exactly at the primes listed in A007352, but earlier for changes to negative sign, and later for the opposite changes. Moreover, a change of sign in either of the sequences corresponds not necessarily to a change of sign (in the strict sense, i.e., regarding 0 as a number with the same sign as the preceding term) in the other one. - M. F. Hasler, Oct 09 2011

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000

Cf. A007352, A112632, A098044, A096630.

Concerning zeros or changes of sign, see also A096449 and A275939.

A173950 := proc(n) if (ithprime(n)+1) mod 6 = 0 then 1; elif (ithprime(n)-1) mod 6 = 0 then -1; else 0 ; end if; end proc:
A174695 := proc(n) add(A173950(i),i=1..n) ; end proc:
seq(A174695(n),n=1..90) ; # R. J. Mathar, Nov 30 2010

Accumulate[Table[Which[Divisible[Prime[n]+1,6],1,Divisible[Prime[n]-1,6],-1,True,0],{n,150}]] (* Harvey P. Dale, Apr 24 2019 *)

s=0;forprime(p=1,999,print1(s+=if(p%3-1,p>3,-1)","))  \\ M. F. Hasler, Oct 09 2011

a(n) = A112632(n)-1. - M. F. Hasler, Oct 09 2011

cp's OEIS Frontend

A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

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A007352 Where the prime race 3k-1 vs. 3k+1 changes leader.

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A096629 Values of n for which {p_3, p_4, ..., p_n} (mod 3) contains equal numbers of 1's and 2's.

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A096449 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is equal to the number of such primes congruent to 2 mod 3.

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A096452 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is one less than the number of such primes congruent to 2 mod 3.

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A297006 Primes p for which pi_{3,2}(p) - pi_{3,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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A297005 Values of n for which pi_{3,2}(p_n) - pi_{3,1}(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).

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A174695 Partial sums of A173950.

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