A039704 -id:A039704 - cp's OEIS frontend

A288915 Run lengths in A039704.

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

Offset: 1

Zak Seidov, Jun 19 2017

nonn

Is the sequence bounded?

On Dickson's conjecture this sequence is unbounded. Records: a(1) = 1, a(9) = 2, a(13) = 3, a(39) = 4, a(180) = 6, a(1348) = 7, a(6698) = 8, a(8156) = 10, a(20230) = 11, a(79011) = 12, a(99250) = 13, a(710895) = 15, a(2421600) = 16, a(7128444) = 17, a(11898707) = 18, a(14368535) = 20, a(21943755) = 22, a(519775979) = 25, a(3111006505) = 27. - Charles R Greathouse IV, Jun 19 2017

Zak Seidov, Table of n, a(n) for n = 1..20000

Cf. A039704, A057620, A057622.

Length /@ Split[Mod[Prime[Range[100]], 6]]

t=1;p=2;forprime(q=3,1e3,if((q-p)%6==0,t++,print1(t", ");t=1);p=q) \\ Charles R Greathouse IV, Jun 19 2017

a(70) corrected by Charles R Greathouse IV, Jun 19 2017

A039703 a(n) = n-th prime modulo 5.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(A049084(A045356(n-1))) = even; a(A049084(A045429(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A010874, A039701, A039702, A039704-A039706, A038194, A007652, A039709-A039715.

[p mod 5: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 5, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 5], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 5] (* Vincenzo Librandi, May 06 2014 *)

primes(105) % 5 \\ Zak Seidov, Apr 09 2013

Sum_k={1..n} a(k) ~ (5/2)*n. - Amiram Eldar, Dec 11 2024

A242119 Primes modulo 18.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 1, 5, 11, 13, 1, 5, 7, 11, 17, 5, 7, 13, 17, 1, 7, 11, 17, 7, 11, 13, 17, 1, 5, 1, 5, 11, 13, 5, 7, 13, 1, 5, 11, 17, 1, 11, 13, 17, 1, 13, 7, 11, 13, 17, 5, 7, 17, 5, 11, 17, 1, 7, 11, 13, 5, 1, 5, 7, 11, 7, 13, 5, 7, 11, 17, 7, 13, 1, 5

Offset: 1

Vincenzo Librandi, May 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. sequences of the type Primes mod k: A039701 (k=3), A039702 (k=4), A039703 (k=5), A039704 (k=6), A039705 (k=7), A039706 (k=8), A038194 (k=9), A007652 (k=10), A039709 (k=11), A039710 (k=12), A039711 (k=13), A039712 (k=14), A039713 (k=15), A039714 (k=16), A039715 (k=17), this sequence (k=18), A033633 (k=19), A242120(k=20), A242121 (k=21), A242122 (k=22), A229786 (k=23), A229787 (k=24), A242123 (k=25), A242124 (k=26), A242125 (k=27), A242126 (k=28), A242127 (k=29), A095959 (k=30), A110923 (k=100).

Magma
```
[p mod(18): p in PrimesUpTo(500)];
```
Mathematica
```
Mod[Prime[Range[100]], 18]
```

[mod(p, 18) for p in primes(500)] # Bruno Berselli, May 05 2014

Sum_{i=1..n} a(i) ~ 9n. The derivation is the same as in the formula in A039715. - Jerzy R Borysowicz, Apr 27 2022

A039705 a(n) = n-th prime modulo 7.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(A049084(A045370(n-1))) is even; a(A049084(A045415(n-1))) is odd. - Reinhard Zumkeller, Feb 25 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A039701-A039704, A039706, A038194, A007652, A039709-A039715.

[p mod(7): p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 7, n=1..105); # Nathaniel Johnston, Jun 29 2011

Table[Mod[Prime[n], 7], {n, 105}] (* Nathaniel Johnston, Jun 29 2011 *)
Mod[Prime[Range[100]], 7] (* Vincenzo Librandi, May 06 2014 *)

primes(100)%7 \\ Charles R Greathouse IV, May 06 2014

Sum_k={1..n} a(k) ~ (7/2)*n. - Amiram Eldar, Dec 11 2024

A099618 a(n) = 1 if the n-th prime == 1 mod 6, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 19 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions.

Characteristic function of A091178.

Cf. A000040, A039704, A132194.

Table[Mod[Mod[Mod[Mod[Prime[k], 6], 5], 3], 2], {k, 1, 120}]
a[n_] := Boole[Mod[Prime[n], 6] == 1]; Array[a, 120] (* Amiram Eldar, Mar 14 2025 *)

a(n) = (prime(n) % 6) == 1; \\ Michel Marcus, Jun 26 2019

From Amiram Eldar, Mar 14 2025: (Start)
a(n) = 1 - A132194(m).
Sum_{k=1..n} a(k) ~ n / 2. (End)

A247478 Primes p such that (p^4 + 5)/6 is prime.

Original entry on oeis.org

7, 11, 17, 29, 53, 71, 101, 109, 127, 179, 227, 241, 281, 307, 349, 487, 587, 647, 683, 727, 829, 1009, 1061, 1109, 1289, 1487, 1511, 1523, 1567, 1621, 1627, 1709, 1847, 1987, 2017, 2027, 2087, 2099, 2297, 2311, 2393, 2437, 2447, 2521, 2531, 2617, 2729, 2887, 2909, 2969, 3167, 3221, 3301, 3319, 3329, 3347, 3413, 3527

Offset: 1

Zak Seidov, Jan 19 2015

nonn

(p^4+5)/6 is an integer for all primes p>3, because then p == (1 or 5) (mod 6) as in A039704, therefore p^2 == 1 (mod 6) and finally p^4 == 1 (mod 6).

			(7^4+5)/6 = 401 prime, (11^4+5)/6 = 2441 prime.

Cf. A118915.

[p: p in PrimesInInterval(3, 4000) | IsPrime((p^4+5) div 6)]; //  Vincenzo Librandi, Jan 21 2015

Select[Prime[Range[10^3]], PrimeQ[(#^4 + 5) / 6] &] (* Vincenzo Librandi, Jan 21 2015 *)

lista(nn) = {forprime(p=4, nn, if (isprime((p^4 + 5)/6), print1(p, ", ")););} \\ Michel Marcus, Jan 20 2015

A079950 Triangle of n-th prime modulo twice primes less n-th prime.

Original entry on oeis.org

2, 3, 3, 1, 5, 5, 3, 1, 7, 7, 3, 5, 1, 11, 11, 1, 1, 3, 13, 13, 13, 1, 5, 7, 3, 17, 17, 17, 3, 1, 9, 5, 19, 19, 19, 19, 3, 5, 3, 9, 1, 23, 23, 23, 23, 1, 5, 9, 1, 7, 3, 29, 29, 29, 29, 3, 1, 1, 3, 9, 5, 31, 31, 31, 31, 31, 1, 1, 7, 9, 15, 11, 3, 37, 37, 37, 37, 37, 1, 5, 1, 13, 19, 15, 7, 3, 41

Offset: 1

Reinhard Zumkeller, Jan 19 2003

The right border of the triangle are the primes: T(n,n)=A000040(n); T(n,1)=A039702(n), T(n,2)=A039704(n) for n>1, T(n,3)=A007652(n) for n>2, T(n,4)=A039712(n) for n>3;

			Triangle begins:
  2;
  3, 3;
  1, 5, 5;
  3, 1, 7,  7;
  3, 5, 1, 11, 11;
  1, 1, 3, 13, 13, 13;
  1, 5, 7,  3, 17, 17, 17;
  ...

Cf. A001747, A079951, A079952, A079953.

A079950 := proc(n,k)
    modp(ithprime(n),2*ithprime(k)) ;
end proc:
seq(seq(A079950(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 28 2017

T(n,k) = prime(n) % (2*prime(k));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Sep 21 2017

T(n, k) = prime(n) mod 2*prime(k), 1<=k<=n.

A181938 Isolated primes = 1 mod 6: sandwiched by primes = 5 mod 6.

Original entry on oeis.org

7, 13, 19, 43, 97, 103, 109, 127, 139, 181, 193, 229, 241, 283, 307, 313, 349, 397, 409, 421, 457, 463, 487, 499, 643, 691, 709, 769, 787, 811, 823, 829, 853, 859, 877, 883, 907, 919, 937, 967, 1021, 1051, 1093, 1153, 1171, 1279, 1303, 1423, 1429, 1447, 1483

Offset: 1

Zak Seidov, Apr 03 2012

nonn

Primes p(m) = 1 mod 6 such that both p(m-1) and p(m+1) are congruent to 5 mod 6.
Corresponding indices m are 4, 6, 8, 14, 25, 27, 29, 31 (A181978).
Note that values of d = p(m+1) - p(m-1) are multiples of 6.

			7 = p(4) = 1 mod 6 and both p(3) = 5 and p(5) = 11 are congruent to 5 mod 6,
13 = p(6) = 1 mod 6 and both p(5) = 11 and p(7) = 17 are congruent to 5 mod 6,
43 = p(14) = 1 mod 6 and both p(13) = 41 and p(15) = 47 are congruent to 5 mod 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002476, A039704, A055625, A181978, A210248.

Select[Prime[Range[2, 300]], Mod[#, 6] == 1 && Mod[NextPrime[#, -1], 6] == 5 && Mod[NextPrime[#, 1], 6] == 5 &] (* T. D. Noe, Apr 04 2012 *)
Transpose[Select[Partition[Prime[Range[250]],3,1],Mod[#[[1]],6] == Mod[#[[3]],6] == 5&&Mod[#[[2]],6]==1&]][[2]] (* Harvey P. Dale, Sep 17 2012 *)

A141455 Irregular triangle showing the set of all possible values of primes modulo n in row n.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 5, 0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 5, 7, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 7, 8, 11, 13, 14, 1, 2, 3, 5, 7, 9, 11, 13, 15

Offset: 2

Roger L. Bagula and Gary W. Adamson, Aug 07 2008

			Table begins:0, 1;
0, 1, 2;
1, 2, 3;
0, 1, 2, 3, 4;
1, 2, 3, 5;
0, 1, 2, 3, 4, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 5, 7, 11;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 5, 7, 9, 11, 13;
1, 2, 3, 4, 5, 7, 8, 11, 13, 14;
1, 2, 3, 5, 7, 9, 11, 13, 15;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
1, 2, 3, 5, 7, 11, 13, 17;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18;
1, 2, 3, 5, 7, 9, 11, 13, 17, 19;

Michael De Vlieger, Table of n, a(n) for n = 2..10209 (rows n = 2..180, flattened)

Cf. A057859 (row lengths), A039701 (row n=3), A039704 (row n=6), A027748, A038566.

Table[Union[FactorInteger[n][[All, 1]] /. n -> 0, Select[Range[n - 1], CoprimeQ[n, #] &]], {n, 2, 15}] (* Michael De Vlieger, Apr 18 2022 *)

Row n = A027748(n) U A038566(n), writing n as 0 iff n is prime. - Michael De Vlieger, Apr 18 2022

A263483 a(n) = prime(n)+(prime(n) modulo 6).

Original entry on oeis.org

4, 6, 10, 8, 16, 14, 22, 20, 28, 34, 32, 38, 46, 44, 52, 58, 64, 62, 68, 76, 74, 80, 88, 94, 98, 106, 104, 112, 110, 118, 128, 136, 142, 140, 154, 152, 158, 164, 172, 178, 184, 182, 196, 194, 202, 200, 212, 224, 232, 230, 238, 244, 242, 256, 262, 268, 274, 272, 278, 286, 284, 298, 308, 316, 314

Offset: 1

Zak Seidov, Oct 19 2015

nonn

For n>2, a(n)-a(n+1)=2 iff prime(n) and prime(n+1) are twin primes; e.g., a(3)-a(4)=10-8=2 and prime(3)=5 and prime(4)=7 are twin primes.

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

Cf. A000040, A039704.

p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  A[n]:= p + (p mod 6);
od:
seq(A[n],n=1..100); # Robert Israel, Jul 18 2018

Mathematica
```
Table[(p=Prime[n])+Mod[p,6],{n,100}]
```

a(n) = apply(x->(x + x%6), prime(n)); \\ Michel Marcus, Oct 27 2015

a(n) = A000040(n) + A039704(n). - Michel Marcus, Oct 27 2015

cp's OEIS Frontend

A288915 Run lengths in A039704.

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A039703 a(n) = n-th prime modulo 5.

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A242119 Primes modulo 18.

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A039705 a(n) = n-th prime modulo 7.

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A099618 a(n) = 1 if the n-th prime == 1 mod 6, otherwise a(n) = 0.

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A247478 Primes p such that (p^4 + 5)/6 is prime.

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Magma

Mathematica

PARI

A079950 Triangle of n-th prime modulo twice primes less n-th prime.

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PARI