A038698 -id:A038698 - cp's OEIS frontend

A007350 Where the prime race 4k-1 vs. 4k+1 changes leader.

3, 26861, 26879, 616841, 617039, 617269, 617471, 617521, 617587, 617689, 617723, 622813, 623387, 623401, 623851, 623933, 624031, 624097, 624191, 624241, 624259, 626929, 626963, 627353, 627391, 627449, 627511, 627733, 627919, 628013, 628427, 628937, 629371

Offset: 1

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

The following references include some on the "prime race" question that are not necessarily related to this particular sequence. - N. J. A. Sloane, May 22 2006

Starting from a(12502) = A051025(27556) = 9103362505801, the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 194367 terms (see a-file) with a(194367) = 9543313015387 as its last term. - Sergei D. Shchebetov, Oct 13 2017

Ford, Kevin; Konyagin, Sergei; Chebyshev's conjecture and the prime number race. IV International Conference "Modern Problems of Number Theory and its Applications": Current Problems, Part II (Russian) (Tula, 2001), 67-91.
Granville, Andrew; Martin, Greg; Prime number races. (Spanish) With appendices by Giuliana Davidoff and Michael Guy. Gac. R. Soc. Mat. Esp. 8 (2005), no. 1, 197-240.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (terms 1..125 from Vincenzo Librandi, terms 126..301 from Robert G. Wilson v)
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.
Andrey Feuerverger and Greg Martin, Biases in the Shanks-Renyi prime number race Experiment. Math. 9 (2000), no. 4, 535-570.
Kevin Ford and Sergei Konyagin, Chebyshev's conjecture and the prime number race, 2002.
Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line, Duke Math. J., Volume 113, Number 2 (2002), 313-330.
Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line. II, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 40 pp., Bonner Math. Schriften, 360, 2003.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Jerzy Kaczorowski, A contribution to the Shanks-Renyi race problem, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 176, 451-458.
Jerzy Kaczorowski, On the Shanks-Renyi race problem mod 5. J. Number Theory 50 (1995), no. 1, 106-118.
Greg Martin, Asymmetries in the Shanks-Renyi prime number race, arXiv:math/0010086 [math.NT], 2000; Number theory for the millennium, II (Urbana, IL, 2000), 403-415, A K Peters, Natick, MA, 2002.
J.-C. Puchta, On large oscillations of the remainder of the prime number theorems Acta Math. Hungar. 87 (2000), no. 3, 213-227.
M. Rubinstein and P. Sarnak, Chebyshev's bias, Exper. Math., 3 (1994), 173-197.
Andrey S. Shchebetov and Sergei D. Shchebetov, First 194367 terms (zipped file).
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
G. M. Ziegler, The great prime number record races, Notices Amer. Math. Soc. 51 (2004), no. 4, 414-416.

Cf. A007350, A007351, A007352, A007353, A007354, A297406, A297407, A297408, A297410, A297411, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A038698 (another way to watch this race).

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

lim = 10^5; k1 = 0; k3 = 0; t = Table[{p = Prime[k], If[Mod[p, 4] == 1, ++k1, k1], If[Mod[p, 4] == 3, ++k3, k3]}, {k, 2, lim}]; A007350 = {3}; Do[ If[t[[k-1, 2]] < t[[k-1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] > t[[k+1, 3]] || t[[k-1, 2]] > t[[k-1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] < t[[k+1, 3]], AppendTo[A007350, t[[k+1, 1]]]], {k, 2, Length[t]-1}]; A007350 (* Jean-François Alcover, Sep 07 2011 *)
lim = 10^5; k1 = 0; k3 = 0; p = 2; t = {}; parity = Mod[p, 4]; Do[p = NextPrime[p]; If[Mod[p, 4] == 1, k1++, k3++]; If[(k1 - k3)*(parity - Mod[p, 4]) > 0, AppendTo[t, p]; parity = Mod[p, 4]], {lim}]; t (* T. D. Noe, Sep 07 2011 *)

A066520 Number of primes of the form 4m+3 <= n minus number of primes of the form 4m+1 <= n.

Original entry on oeis.org

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

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jan 05 2002

Although the initial terms are nonnegative, it has been proved that infinitely many terms are negative. The first two are a(26861)=a(26862)=-1. Next there are 3404 values of n in the range 616841 to 633798 with a(n)<0. Then 27218 values in the range 12306137 to 12382326.

Partial sums of A151763. - Reinhard Zumkeller, Feb 06 2014

T. D. Noe, Table of n, a(n) for n=1..30000 (enough terms to show the first dip into negative territory)
Carter Bays and Richard H. Hudson, Zeros of Dirichlet L-Functions and Irregularities in the Distribution of Primes, Mathematics of Computation, 69 (2000) 861-866.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Cf. A066339, A066490, A007350, A051024, A051025.
Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]
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), this sequence (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).

a066520 n = a066520_list !! (n-1)
a066520_list = scanl1 (+) $ map (negate . a151763) [1..]
-- Reinhard Zumkeller, Feb 06 2014

a[n_] := Length[Select[Range[3, n, 4], PrimeQ]]-Length[Select[Range[1, n, 4], PrimeQ]]
f[n_]:=Module[{c=Mod[n,4]},Which[!PrimeQ[n],0,c==3,1,c==1,-1]]; Join[{0,0}, Accumulate[Array[f,110,3]]] (* Harvey P. Dale, Mar 03 2013 *)

a(n) = A066490(n) - A066339(n).

a(2*n+1) = a(2*n+2) = -A156749(n). - Jonathan Sondow, May 17 2013

Edited by Dean Hickerson, Mar 05 2002

A038691 Indices of primes at which the prime race 4k-1 vs. 4k+1 is tied.

Original entry on oeis.org

1, 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, 50375, 50377, 50379, 50381, 50393, 50413, 50423, 50425, 50427, 50429, 50431, 50433, 50435, 50437, 50439, 50445, 50449, 50451, 50503, 50507, 50515, 50517, 50821, 50843, 50853, 50855, 50857, 50859, 50861

Offset: 1

Hans Havermann

nonn

Starting from a(27410) = 316064952537 the sequence includes the 8th sign-changing zone predicted by C. Bays et al back in 2001. The sequence with the first 8 sign-changing zones contains 419467 terms (see a-file) with a(419467) = 330797040309 as its last term. - Sergei D. Shchebetov, Oct 16 2017

			From _Jon E. Schoenfield_, Jul 24 2021: (Start)
a(n) is the n-th number m at which the prime race 4k-1 vs. 4k+1 is tied:
.
                             count
                           ----------
   m  p=prime(m)  p mod 4  4k-1  4k+1
  --  ----------  -------  ----  ----
   1       2         2       0  =  0    a(1)=1
   2       3        -1       1     0
   3       5        +1       1  =  1    a(2)=3
   4       7        -1       2     1
   5      11        -1       3     1
   6      13        +1       3     2
   7      17        +1       3  =  3    a(3)=7
   8      19        -1       4     3
   9      23        -1       5     3
  10      29        +1       5     4
  11      31        -1       6     4
  12      37        +1       6     5
  13      41        +1       6  =  6    a(4)=13
(End)

Stan Wagon, The Power of Visualization, Front Range Press, 1994, pp. 2-3.

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
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, pp. 173-197.
Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..419647 (zipped file)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A002145, A002313, A007350, A007351, A038698, A051024, A051025, A066520, A096628, A096447, A096448, A199547.

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

Flatten[ Position[ FoldList[ Plus, 0, Mod[ Prime[ Range[ 2, 50900 ] ], 4 ]-2 ], 0 ] ]

lista(nn) = {nbp = 0; nbm = 0; forprime(p=2, nn, if (((p-1) % 4) == 0, nbp++, if (((p+1) % 4) == 0, nbm++)); if (nbm == nbp, print1(primepi(p), ", ")););} \\ Michel Marcus, Nov 20 2016

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

Original entry on oeis.org

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

A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 60, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 120, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 150, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 180, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212

Offset: 1

Amiram Eldar, Feb 26 2021

Numbers k such that A276084(k) is odd.
All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * Sum_{k=1..m}(-1)^k/A002110(k) = 1, 2, 11, 76, 837, 10880, 184961, ...
The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/A002110(k) = 0.362306... (A132120).
Also Heinz numbers of partitions with even least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021
Numbers k such that A000720(A053669(k)) is even. Differences from the related A353531 seem to be terms that are multiples of 210, but not all of them, for example primorial 30030 (= 143*210) is in neither sequence. Consider also A038698. - Antti Karttunen, Apr 25 2022

			2 is a term since A049345(2) = 10 has 1 trailing zero.
4 is a term since A049345(2) = 20 has 1 trailing zero.
30 is a term since A049345(2) = 1000 has 3 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}             46: {1,9}             90: {1,2,2,3}
      4: {1,1}           50: {1,3,3}           92: {1,1,9}
      8: {1,1,1}         52: {1,1,6}           94: {1,15}
     10: {1,3}           56: {1,1,1,4}         98: {1,4,4}
     14: {1,4}           58: {1,10}           100: {1,1,3,3}
     16: {1,1,1,1}       60: {1,1,2,3}        104: {1,1,1,6}
     20: {1,1,3}         62: {1,11}           106: {1,16}
     22: {1,5}           64: {1,1,1,1,1,1}    110: {1,3,5}
     26: {1,6}           68: {1,1,7}          112: {1,1,1,1,4}
     28: {1,1,4}         70: {1,3,4}          116: {1,1,10}
     30: {1,2,3}         74: {1,12}           118: {1,17}
     32: {1,1,1,1,1}     76: {1,1,8}          120: {1,1,1,2,3}
     34: {1,7}           80: {1,1,1,1,3}      122: {1,18}
     38: {1,8}           82: {1,13}           124: {1,1,11}
     40: {1,1,1,3}       86: {1,14}           128: {1,1,1,1,1,1,1}
     44: {1,1,5}         88: {1,1,1,5}        130: {1,3,6}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A002110, A049345, A053669, A132120, A276084.
Complement of A342051.
A099800 is subsequence.
Analogous sequences: A001950 (Zeckendorf representation), A036554 (binary), A145204 (ternary), A217319 (base 4), A232745 (factorial base).
The version for reversed binary expansion is A079523.
Positions of even terms in A257993.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339662 gives greatest gap in prime indices.
Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function).
Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.

seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100],EvenQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)

A353525(n) = { for(i=1,oo,if(n%prime(i),return((i+1)%2))); }
isA342050(n) = A353525(n);
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n,", "))); \\ Antti Karttunen, Apr 25 2022

More terms added (to differentiate from A353531) by Antti Karttunen, Apr 25 2022

A321856 Number of primes of the form 3m + 2 <= n minus number of primes of the form 3m + 1 <= n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 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 3 minus the number of primes <= n that are quadratic residues modulo 3.
Conjecturally infinitely many terms are negative. The earliest negative term is a(608981813029) = -1, see A112632.
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 05 2023]

			Below 100, there are 11 primes congruent to 1 modulo 3 and 13 primes congruent to 2 modulo 3, so a(100) = 13 - 11 = 2.

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

Cf. A007350, A102283, A340763, A340764.
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), this sequence (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(-3, i))

a(n) = -Sum_{primes p<=n} Legendre(p,3) = -Sum_{primes p<=n} Kronecker(-3,p) = -Sum_{primes p<=n} A102283(p).

a(n) = A340764(n) - A340763(n). - Jianing Song, May 06 2021

A071838 a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(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, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2

Offset: 1

Benoit Cloitre, Jun 08 2002

a(n) is the number of odd primes <= n that have 2 as a quadratic nonresidue minus the number of primes <= n that have 2 as a quadratic residue. See the comments about "Chebyshev's bias" in A321861. - Jianing Song, Nov 24 2018
Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1?
The first negative term occurs at a(11100143) = -1. - Jianing Song, Nov 08 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Wikipedia, Chebyshev's bias

Cf. 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), this sequence (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).

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* Michael De Vlieger, Nov 25 2018 *)

for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-3)%8,0,1)+if((i*isprime(i)-5)%8,0,1)-if((i*isprime(i)-1)%8,0,1)-if((i*isprime(i)-7)%8,0,1)),", ")) \\ Program fixed by Jianing Song, Nov 08 2019

a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ Jianing Song, Nov 24 2018

a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - Jianing Song, Nov 20 2018

Edited by Peter Munn, Nov 19 2023

A320857 a(n) = Pi(8,5)(n) + Pi(8,7)(n) - Pi(8,1)(n) - Pi(8,3)(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, -1, -1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2

Offset: 1

Jianing Song, Nov 24 2018

sign

a(n) is the number of odd primes <= n that have -2 as a quadratic nonresidue minus the number of primes <= n that have -2 as a quadratic residue.
It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 212 ones are 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 18 2023]
Here, although 3 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).

			Pi(8,1)(200) = 8, Pi(8,5)(200) = 13, Pi(8,3)(200) = Pi(8,7)(200) = 12, so a(200) = 13 + 12 - 8 - 12 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias.

Cf. A007350, A188510.
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), this sequence (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).

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[-2, #], 0] &, 88] (* Michael De Vlieger, Nov 25 2018 *)

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

from sympy import isprime; from numpy import sign
def A320857(n): return sum(isprime(i)*(i%2)*sign(i%8-4) for i in range(1,n+1)) # Ya-Ping Lu, Jan 25 2025

a(n) = -Sum_{primes p<=n} Kronecker(-2,p) = -Sum_{primes p<=n} A188510(p).

A320858 a(n) = A320857(prime(n)).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 24 2018

sign

Among the first 10000 terms there are only 100 negative ones. See the comments about "Chebyshev's bias" in A320857.

			prime(46) = 199, Pi(8,1)(199) = 8, Pi(8,5)(199) = 13, Pi(8,3)(199) = Pi(8,7)(199) = 12, so a(46) = 13 + 12 - 8 - 12 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Chebyshev's bias.

Cf. A188510.
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), this sequence (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a[n_] := -Sum[KroneckerSymbol[-2, Prime[i]], {i, 1, n}];
Array[a, 100] (* Jean-François Alcover, Dec 28 2018, from PARI *)

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)).

A321857 a(n) = Pi(5,2)(n) + Pi(5,3)(n) - Pi(5,1)(n) - Pi(5,4)(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, 1, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of primes <= n that are quadratic nonresidues modulo 5 minus the number of primes <= n that are quadratic residues modulo 5.
a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative.
The first negative term occurs at a(2082927221) = -1. - Jianing Song, Nov 08 2019
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			Pi(5,1)(100) = Pi(5,4)(100) = 5, Pi(5,2)(100) = Pi(5,3)(100) = 7, so a(100) = 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), this sequence (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(5, i))

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

Edited by Peter Munn, Nov 18 2023

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A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

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A321856 Number of primes of the form 3*m + 2 <= n minus number of primes of the form 3*m + 1 <= n.

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A321856 Number of primes of the form 3m + 2 <= n minus number of primes of the form 3m + 1 <= n.