A038691 -id:A038691 - cp's OEIS frontend

A093180 Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

26829, 26845, 26853, 26857, 26865, 26869, 26873, 26877, 26885, 26889, 26897, 26917, 616765, 616773, 616777, 616785, 616825, 616833, 616837, 616845, 616853, 616857, 616861, 616865, 616869, 616873, 616881, 616885, 616889, 616893, 617013

Offset: 1

Enoch Haga, Mar 27 2004

This odd composite race seems analogous to the prime number race

			a(1)=26829 is the first odd composite 1 mod 4 where the count 5238 is the same for 26835 3 mod 4

Cf. A007350, A007351, A038691, A093181, A093182.

Beginning with 1, run separate counts for odd composites 1 mod 4 and 3 mod 4. When the count is equal, record the number for 1 mod 4.

A093181 Odd composites (including 1 in the count) where the number 3 mod 4 equals the number 1 mod 4.

Original entry on oeis.org

26835, 26851, 26855, 26859, 26867, 26871, 26875, 26883, 26887, 26895, 26899, 26923, 616771, 616775, 616779, 616795, 616831, 616835, 616839, 616847, 616851, 616855, 616859, 616863, 616867, 616875, 616879, 616883, 616887, 616891, 617007

Offset: 1

Enoch Haga, Mar 27 2004

This odd composite race seems analogous to the prime number race

			a(1)=26835 is the first odd composite 3 mod 4 where the count 5238 is the same for 26829 1 mod 4

Cf. A007350, A007351, A038691, A093180, A093182.

Beginning with 1, run separate counts for odd composites 1 mod 4 and 3 mod 4. When the count is equal, record the number for 3 mod 4.

A093182 Counts where both the odd composites (starting from 1) 1 mod 4 and 3 mod 4 are equal.

Original entry on oeis.org

5238, 5241, 5242, 5243, 5244, 5245, 5246, 5247, 5248, 5249, 5250, 5255, 129008, 129009, 129010, 129012, 129020, 129021, 129022, 129023, 129024, 129025, 129026, 129027, 129028, 129029, 129030, 129031, 129032, 129033, 129058, 129059, 129060

Offset: 1

Enoch Haga, Mar 27 2004

			At 26829 1 mod 4 and 26835 3 mod 4, the count of odd composites is equal for each run at 5238; so a(1)=5238. [Compare to the prime 26833 1 mod 4 where equality occurs at count 1471 and the first reversal in the race occurs at 26861.]

Cf. A093180, A093181, A007350, A007351, A038691.

Run separate counts of odd composites 1 mod 4 and 3 mod 4. When the count is equal, record the count.

A216057 a(n) = A045429(n) - A045356(n).

Original entry on oeis.org

1, 6, 6, 6, 12, 12, 6, 6, 2, 4, -6, -6, 4, -4, 4, 4, 14, 24, 24, 24, 24, 14, 14, 24, 6, 12, 12, 6, 2, 4, -24, -24, -26, -34, -18, -6, 6, 4, 12, 24, 22, 14, 4, 12, 6, 24, 24, 34, 24, 32, 16, 14, 24, 24, 26, 32, 34, 26, 34, 14, 14, 6, -18, -18, 6, 4, -6, -8, -14

Offset: 1

Zak Seidov, Aug 31 2012

sign

Zak Seidov, Table of n, a(n) for n = 1..1000
Zak Seidov, Plot of terms up to a(1000) (gif image)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319v1 [math.NT]
Eric Weisstein's World of Mathematics, Chebyshev Bias

Cf. A007350, A007351, A038691, A038698, A045356, A045429, A156749.

A295353 Values of n for which pi_{8,7}(p_n) - pi_{8,1}(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

6035005477560, 6035005477596, 6035005477608, 6035005477618, 6035005477620, 6035005477623, 6035005477632, 6035005478719, 6035005478725, 6035005478730, 6035005478822, 6035005478826, 6035005478829, 6035005478863, 6035005478866, 6035005478874, 6035005479026, 6035005479132, 6035005479158, 6035005479163

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Nov 20 2017

nonn

This sequence is a companion sequence to A295354. The sequence with the first found pi_{8,7}(p_n) - pi_{8,1}(p_n) sign-changing zone contains 234937 terms (see a-file) with a(237937) = 6053968231350 as its last term.

Andrey S. Shchebetov and 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.
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.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547, A275939, A295354

A297354 Values of n for which pi_{12,5}(p_n) - pi_{12,1}(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

862062606318, 862062606330, 862062606348, 862062606351, 862062606377, 862062606380, 862062606387, 862062606393, 862062606424, 862062606448, 862062606453, 862062606466, 862062606469, 862062606478, 862062606481, 862062606488, 862062606490, 862062606494, 862062606496, 862062606500

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 29 2017

nonn

This is a companion sequence to A297355 and includes values of n for the first discovered sign-changing zone for pi_{12,5}(p) - pi_{12,1}(p) prime race. The full sequence checked up to 10^14 has 8399 terms (see b-file).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..8399
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.
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. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A297355.

A297355 Primes p for which pi_{12,5}(p) - pi_{12,1}(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

25726067172577, 25726067172857, 25726067173321, 25726067173441, 25726067174389, 25726067174461, 25726067174653, 25726067174761, 25726067175961, 25726067176549, 25726067176669, 25726067176993, 25726067177149, 25726067177429, 25726067177449, 25726067177593, 25726067177617, 25726067177689, 25726067177801, 25726067178013

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 29 2017

nonn

This is a companion sequence to A297354 and includes the first discovered sign-changing zone for pi_{12,5}(p) - pi_{12,1}(p) prime race. The full sequence checked up to 10^14 has 8399 terms (see b-file).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..8399
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.
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. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A297354.

A330359 Race of lucky numbers of the form 4k - 1 vs. 4k + 1 is tied at the a(n)-th lucky number.

Original entry on oeis.org

2, 4, 6, 16, 20, 22, 24, 2684, 2686, 2688, 2696, 2710, 2712, 109978, 110026, 110028, 110030, 110052, 110056, 110060, 110068, 110070, 110154, 110156, 110158, 110160, 118048, 118050, 118126, 118128, 118130, 118132, 118134, 118136, 118138, 118152, 118154, 118156

Offset: 1

Amiram Eldar, Dec 12 2019

nonn

All the terms are even by definition. For each term m, there are m/2 lucky numbers of the form 4*k - 1 and m/2 lucky numbers of the form 4*k + 1 up to the m-th lucky number.

Gardiner et al. (1956) noted that the ratio between the numbers of lucky numbers of the form 4*k - 1 and 4*k + 1 seems to tend to 1, with a preponderance, at first, of the lucky numbers of the form 4*k + 3.

			6 is in the sequence since the first 6 lucky numbers are 1, 3, 7, 9, 13, 15, half of them are of the form 4*k-1 (3, 7, 15) and half of the form 4*k+1 (1, 9, 13).

Vema Gardiner, Roger Lazarus, Nicholas Metropolis and Stanislaw Ulam, On certain sequences of integers defined by sieves, Mathematics Magazine, Vol. 29, No. 3 (1956), pp. 117-122. See Table IV, p. 121.

Cf. A000959, A038691, A137168, A137170, A330360, A330361.

lucky = Import["b000959.txt", "Table"][[;; , 2]]; Flatten[Position[Accumulate[ Mod[lucky, 4] - 2], 0]] (* use the b-file from A000959 *)

A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

Original entry on oeis.org

2, 3, 11, 71, 83, 223, 227, 503, 751, 1531, 1543, 1571, 1579, 1583, 4127, 5147, 5171, 5179, 5651, 6211, 11083, 11087, 11471, 11483, 11519, 11527, 11579, 11587, 17239, 17551, 17903, 17971, 35963, 36011, 39703, 39727

Offset: 1

Gerhard Kirchner, Nov 20 2021

nonn

The difference d(x) = pi(x,4,3) - pi(x,4,1) changes sign infinitely often, see link "Prime Quadratic Effect". But this does not say anything about the amplitudes of these oscillations. For diagrams, see link "Oscillations of d(x)". If d(x) has no upper limit, the current sequence is infinite. Regarding the lower limit, see A349519.

			primes 4*j+1: 5, 13, 17, ...
       4*j+3: 3, 7, 11, ...
d(x) = pi(x,4,3) - pi(x,4,1)
.
  n  x  pi(x,4,3) pi(x,4,1)  d(x)=n-1?
  - --  --------- ---------  ---------
  1  2       0         0     0=0  true   a(1)  = 2
  2  3       1         0     1=1  true   a(2)  = 3
  3  5       1         1     0=2 false   a(3) != 5
  ...........................
  3 11       3         1     2=2  true   a(3)  = 11

Gerhard Kirchner, Oscillations of d(x)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A038691, A007350.

block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 40,
  /* returns nmax terms */
while nsum then(n:n+1, sum:su, w: append(w,[p]) ) ) ,
w);

A118480 (n-th 4k+1 prime minus n-th 4k+3 prime minus two)/4.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 2, 3, 3, 5, 6, 5, 6, 2, 7, 5, 6, 8, 7, 7, 7, 12, 10, 10, 11, 11, 12, 10, 10, 12, 11, 13, 10, 10, 10, 10, 9, 8, 7, 9, 3, 4, 4, 4, 11, 13, 15, 17, 19, 19, 22, 19, 16, 13, 17, 16, 15, 16, 14, 17, 16, 21, 24, 19, 19, 13, 17, 17, 19, 19, 16, 11, 13, 13, 22, 19, 19, 17, 22

Offset: 1

Clark Kimberling and Robert G. Wilson v, May 04 2006

sign

Zero occurs infinitely often as do the negative numbers.
First occurrence of a(n) beginning with 0: 1, 2, 4, 5, 42, 10, 11, 15, 18, 37, 23, 25, 22, 32, 59, 47, 53, 48, 83, 49, 110, 62, 51, 82, 63, 170, ...,
The first negative term is at n=1473. - T. D. Noe, Apr 09 2009

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A022757, A007350, A007351, A038691.

(Select[1 + 4Range@245, PrimeQ@# &] - Select[ -1 + 4Range@225, PrimeQ@# &] - 2)/4

a(n) = (A002144(n) - A002145(n) - 2)/4.

cp's OEIS Frontend

A093180 Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

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A093181 Odd composites (including 1 in the count) where the number 3 mod 4 equals the number 1 mod 4.

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A093182 Counts where both the odd composites (starting from 1) 1 mod 4 and 3 mod 4 are equal.

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A216057 a(n) = A045429(n) - A045356(n).

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A295353 Values of n for which pi_{8,7}(p_n) - pi_{8,1}(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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A297354 Values of n for which pi_{12,5}(p_n) - pi_{12,1}(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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A297355 Primes p for which pi_{12,5}(p) - pi_{12,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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A330359 Race of lucky numbers of the form 4*k - 1 vs. 4*k + 1 is tied at the a(n)-th lucky number.

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Mathematica

A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

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Maxima

A118480 (n-th 4k+1 prime minus n-th 4k+3 prime minus two)/4.

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Mathematica

A330359 Race of lucky numbers of the form 4k - 1 vs. 4k + 1 is tied at the a(n)-th lucky number.