A096630 -id:A096630 - cp's OEIS frontend

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

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

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

A306891 Primes p for which pi_{3,2}(p) < pi_{3,1}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Original entry on oeis.org

608981813029, 608981813137, 608981813191, 608981813261, 608981813269, 608981813273, 608981813311, 608981813347, 608981813357, 608981813449, 608981813459, 608981813683, 608981813701, 608981813707, 608981813711, 608981813717, 608981813719, 608981813777, 608981813779

Offset: 1

Jianing Song, Mar 16 2019

nonn

Primes p such that Sum_{primes q <= p} Kronecker(-3,q) > 0.

Indices of negative terms in A321856. See also the comment about Chebyshev's bias in A321856.

David A. Corneth, Table of n, a(n) for n = 1..10000
C. Bays, R. H. Hudson, Details of the First Region of Integers x with pi_{3,2}(x) < pi_{3,1}(x), Mathematics of Computation 32(142), 1978, pp. 571-576.
Eric Weisstein's World of Mathematics, Chebyshev Bias
Wikipedia, Chebyshev's bias

Cf. A096630, A112632, A297006, A321856.

Cf. also A199547, A096628.

my(i=0); forprime(p=608981813029, 608981820000, i+=kronecker(-3, p); if(i>0, print1(p, ", ")))

a(n) = prime(A096630(n)).

cp's OEIS Frontend

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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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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A306891 Primes p for which pi_{3,2}(p) < pi_{3,1}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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