A039702 -id:A039702 - cp's OEIS frontend

A092663 Number of consecutive prime runs of 10 primes congruent to 1 mod 4 below 10^n.

0, 0, 0, 0, 0, 5, 19, 323, 3653, 37544, 381413, 3799344, 37591054

Offset: 1

Enoch Haga, Mar 02 2004

			a(6)=5 because 5 sets of 10 primes occur below 10^6, each run interrupted by a prime congruent to 3 mod 4. These runs start at prime(31798)=373649, prime(41181)=495377, at prime(42241)=509389, at prime(50017)=612109, *not* at prime(61457) which has a larger run length, and at prime(63146)=789097.

Lucas A. Brown, Python program.

Cf. A092664, A092665, A039702, A100672, A145990.

Generate the prime sequence with primes labeled 1 mod 4 or 3 mod 4. Add count of primes to sequence if just 10 primes occur before interruption by a prime congruent to 3 mod 4

a(9) and a(10) from Sean A. Irvine, Oct 06 2011
a(11) from Chai Wah Wu, Mar 18 2018
a(12) and a(13) from Lucas A. Brown, Oct 15 2024

A103271 a(n) = (prime(n)+prime(n+1)) mod 4.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Yasutoshi Kohmoto, Jan 27 2005

nonn

The number of 2's among the first N terms are: count(10^3) = 381, count (10^4) = 4137, count(10^5) = 42638, count(10^6) = 437423, count(10^7) = 4448503. - M. F. Hasler, Apr 27 2016

In terms of vectors a = (p(n),p(n+1)) mod 4, as considered in the preprint arxiv:1603.03720, the 2's group together the cases a = (1,1) and (3,3) and 0's cumulate cases (1,3) and (3,1). Assuming that the two subcases of each case have roughly the same probabilities, the above counts (i.e., percentage of 44.5% : 55.5% at 10^7) are compatible with the data in the 2nd table on bottom of p.14 where respective percentages vary from 44.8% : 55.1% (at 10^10) to 46% : 54% (at 10^12). I found that at p(n) ~ 10^80, the percentages become closer than 49% : 51%. - M. F. Hasler, May 12 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.

Cf. A001043, A039702.

seq(ithprime(n)+ithprime(n+1) mod 4, n=1..150); # Emeric Deutsch, May 31 2005

Table[Mod[Prime@ n + Prime[n + 1], 4], {n, 120}] (* Michael De Vlieger, Apr 27 2016 *)
Mod[Total[#],4]&/@Partition[Prime[Range[120]],2,1] (* Harvey P. Dale, Mar 16 2025 *)

a(n) = (prime(n) + prime(n+1)) % 4; \\ Michel Marcus, Apr 14 2016

a(n) = A001043(n) mod 4. - Michel Marcus, Apr 14 2016

More terms from Emeric Deutsch, May 31 2005

Prepended a(1) = 1, Joerg Arndt, Apr 14 2016

A271974 Let p = prime(n): if p mod 4 == 1 then a(n) = (1+p)/2 otherwise if p mod 4 == 3 then a(n) = (1-p)/2.

Original entry on oeis.org

-1, 3, -3, -5, 7, 9, -9, -11, 15, -15, 19, 21, -21, -23, 27, -29, 31, -33, -35, 37, -39, -41, 45, 49, 51, -51, -53, 55, 57, -63, -65, 69, -69, 75, -75, 79, -81, -83, 87, -89, 91, -95, 97, 99, -99, -105, -111, -113, 115, 117, -119, 121, -125, 129, -131, 135, -135, 139, 141, -141, 147, -153, -155, 157, 159, -165, 169, -173, 175, 177, -179, -183, 187, -189, -191, 195, 199, 201, 205

Offset: 2

Dimitris Valianatos, Apr 23 2016

sign

			For n=11, prime(11) = 31, 31 mod 4 == 3 so a(11) = (1-31)/2 = -15.

Dimitris Valianatos, Comments on this sequence, Apr 25 2016

Cf. A039702, A272295.

If[Mod[#,4]==1,(1+#)/2,(1-#)/2]&/@Prime[Range[2,80]] (* Harvey P. Dale, May 09 2017 *)

{forstep(n=3,1000,2,if(isprime(n),if(n%4==1,p=(1+n)/2,p=(1-n)/2);print1(n"-> "p", ")));}

Product_{n>2} (1-1/a(n)) = (1-1/3)*(1-1/(-3))*(1-1/(-5))*(1-1/7)*(1-1/9)*(1-1/(-9))*(1-1/(-11))*(1-1/15)*(1-1/(-15))*... = (2/3)*(4/3)*(6/5)*(6/7)*(8/9)*(10/9)*(12/11)*(14/15)*(16/15)*... = 1.
So Product_{n>2} (1-a(n)^(-1)) = Product_{n>2}(1-a(n)^(-1))^(-1) = (Product_{n>2}(1-a(n)^(-1)))^k = 1, for every k.
Sum_ {n>2} log(1-1/a(n)) = 0.

Corrected and extended by Harvey P. Dale, May 09 2017

A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

Original entry on oeis.org

4, 3, 3, 4, 5, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 5, 2, 4, 5, 2, 4, 5, 3, 2, 3, 4, 5, 2, 3, 4, 5, 3, 4, 3, 4, 2, 4, 5, 3, 5, 2, 5, 2, 3, 4, 4, 4, 5, 2, 3, 5, 2, 5, 3, 5, 3, 4, 2, 3, 4, 3, 4, 5, 2, 3, 4, 2, 5, 2, 3, 5, 4, 2, 4, 5, 3, 2, 3, 2, 5, 2, 5, 2, 4, 5, 3, 2, 3, 4, 5, 5, 4, 5, 4, 5, 3, 3, 4, 2, 4, 3

Offset: 1

Zak Seidov, Jan 16 2011

nonn

a(n) = 2 iff prime(n) == 1 (mod 12); a(n) = 2 for prime(n) = 13, 37, 61, 73, 97, 109, ... (A068228).
a(n) = 5 iff prime(n) == 11 (mod 12); a(n) = 5 for prime(n) = 11, 23, 47, 59, 71, 83, ... (A068231).
For n > 2, a(n) = 3 iff prime(n) == 5 (mod 12); a(n) = 3 for prime(n) = 5, 17, 29, 41, 53, 89, ... (A040117).
For n > 2, a(n) = 4 iff prime(n) == 7 (mod 12); a(n) = 4 for prime(n) = 7, 19, 31, 43, 67, 79, ... (A068229).

Equals A039701 + A039702.

Cf. A068228, A068231, A040117, A068229, A039710.

A180217:=func< n | p mod 3 + p mod 4 where p is NthPrime(n) >; [ A180217(n): n in [1..105] ]; // Klaus Brockhaus, Jan 18 2011

Mod[#,3]+Mod[#,4]&/@Prime[Range[110]] (* Harvey P. Dale, Nov 09 2011 *)

A267573 a(n) = prime(n) + (prime(n) mod 4).

Original entry on oeis.org

4, 6, 6, 10, 14, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 62, 62, 70, 74, 74, 82, 86, 90, 98, 102, 106, 110, 110, 114, 130, 134, 138, 142, 150, 154, 158, 166, 170, 174, 182, 182, 194, 194, 198, 202, 214, 226, 230, 230, 234, 242, 242, 254, 258, 266, 270

Offset: 1

Emre APARI, Jan 17 2016

The primes corresponding to the cases where a(n) = a(n+1) can be found in A071698. - Michel Marcus, Jan 17 2016

			p=19; 19 + (19 modulo 4) = 22.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A039702, A071698, A083220.

[NthPrime(n)+(NthPrime(n) mod 4): n in [1..100]]; // Vincenzo Librandi, Jan 17 2016

A267573:=n->ithprime(n)+(ithprime(n) mod 4): seq(A267573(n), n=1..100); # Wesley Ivan Hurt, Jan 17 2016

Table[Prime[n] + Mod[Prime[n], 4], {n, 60}] (* Vincenzo Librandi, Jan 17 2016 *)
#+Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Jun 12 2020 *)

a(n) = prime(n) + (prime(n) % 4); \\ Michel Marcus, Jan 17 2016

lista(nn) = forprime(p=2, nn, print1(p + p % 4, ", ")); \\ Altug Alkan, Jan 17 2016

a(n) = A000040(n) + A039702(n).

a(n) = A083220(prime(n)). - Michel Marcus, Jan 17 2016

More terms from Vincenzo Librandi, Jan 17 2016

A077008 Legendre symbol (-1,p) where p is the n-th prime.

Original entry on oeis.org

-1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1

Offset: 2

Joseph L. Pe, Nov 28 2002

Same as A070750 (where additionally a(1)=0 is included). - Joerg Arndt, Apr 21 2016

Cf. A070750, A039702.

p = Table[Prime[i], {i, 2, 101}]; Table[JacobiSymbol[p[[j]] - 1, p[[j]]], {j, 1, 100}]
a[n_] := (-1)^((Prime[n]-1)/2); Table[a[n], {n, 2, 101}] (* Jean-François Alcover, Apr 20 2016 *)

a(n)=kronecker(-1,prime(n)) \\ Charles R Greathouse IV, Jun 13 2013

a(n)=if(prime(n)%4==1,1,-1) \\ Charles R Greathouse IV, Jun 13 2013

a(n) = A070750(n). - Joerg Arndt, Apr 21 2016

Replaced "(p-1,p)" by "(-1,p)" in name, Joerg Arndt, Apr 21 2016

A202025 Position of second appearance of set of first n terms in the sequence of odd primes modulo 4.

Original entry on oeis.org

3, 4, 8, 16, 16, 19, 60, 221, 654, 654, 654, 654, 654, 30291, 30291, 30291, 30291, 250231, 342916, 342916, 472727, 1934365, 1934365, 11877702, 11877702, 11877702

Offset: 1

Zak Seidov, Dec 09 2011

nonn

Next term, a(27) > 3*10^7.

			Consider the sequence of odd primes modulo 4: S= 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1,... . Then
a(1)=3 because 2nd appearance of 3 is S(3),
a(2)=4 because 2nd appearance of (3,1) begins at S(4),
a(3)=8 because 2nd appearance of (3,1,3) begins at S(8),
a(4)=16 because 2nd appearance of (3,1,3,3) begins at S(16).

Cf. A039702.

nn=3*10^7; s=Table[Mod[Prime[n],4], {n,2,nn}]; Reap[k1=2; Do[tn=Take[s,n]; Do[If[tn==Take[s,{k,k+n-1}], Sow[k]; k1=k; Break[]], {k,k1,nn-n-1}], {n,26}]][[2,1]]

cp's OEIS Frontend

A092663 Number of consecutive prime runs of 10 primes congruent to 1 mod 4 below 10^n.

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A103271 a(n) = (prime(n)+prime(n+1)) mod 4.

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Maple

Mathematica

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A271974 Let p = prime(n): if p mod 4 == 1 then a(n) = (1+p)/2 otherwise if p mod 4 == 3 then a(n) = (1-p)/2.

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A180217 a(n) = (n-th prime modulo 3) + (n-th prime modulo 4).

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Magma

Mathematica

A267573 a(n) = prime(n) + (prime(n) mod 4).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A077008 Legendre symbol (-1,p) where p is the n-th prime.

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Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A202025 Position of second appearance of set of first n terms in the sequence of odd primes modulo 4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs