A024898 -id:A024898 - cp's OEIS frontend

A167055 Numbers k such that 12*k + 5 is prime.

0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21, 22, 23, 24, 26, 29, 32, 33, 37, 38, 42, 43, 46, 47, 49, 51, 53, 54, 56, 58, 63, 64, 66, 67, 68, 71, 73, 77, 78, 79, 81, 84, 87, 88, 91, 92, 98, 99, 101, 102, 106, 107, 108, 113, 114, 117, 119, 123, 124, 129, 133, 134, 136, 141

Offset: 1

Michael B. Porter, Oct 27 2009

Corresponds to odd numbers in A024898.

			2 is in the sequence since 12*2+5 = 29 is prime.

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

Cf. A110801, A167056, A167057, A024898, primes are in A040117.

[n: n in [1..150] | IsPrime(12*n+5)]; // Vincenzo Librandi, May 20 2014

Select[Range[0,150],PrimeQ[12#+5]&] (* Harvey P. Dale, Oct 07 2012 *)

PARI
```
isA167055(n) = isprime(12*n+5)
```

A167057 Numbers k such that 12*k + 11 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 10, 13, 14, 15, 18, 19, 20, 21, 25, 28, 29, 31, 34, 35, 36, 38, 39, 40, 41, 46, 48, 49, 53, 54, 56, 59, 61, 68, 69, 71, 73, 75, 78, 80, 81, 84, 85, 90, 91, 95, 96, 98, 101, 104, 106, 108, 109, 113, 118, 119, 120, 123, 124, 125, 126, 129, 130, 131, 133

Offset: 1

Michael B. Porter, Oct 27 2009

Corresponds to even numbers in A024898.

			3 is in the sequence since 12*3+11 = 47 is prime.

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

Cf. A110801, A167055, A167056, A024898, primes are in A068231.

[n: n in [0..200] |IsPrime(12*n+11)]; // Vincenzo Librandi, Mar 25 2010

Select[Range[0, 200], PrimeQ[12 # + 11] &] (* Vincenzo Librandi, May 20 2014 *)

PARI
```
isA167057(n) = isprime(12*n+11)
```

a(n) = A138620(n)-1. [From R. J. Mathar, Oct 29 2009]

A153134 Numbers k such that 6k - 7 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 19, 20, 23, 24, 26, 29, 30, 31, 33, 34, 39, 40, 41, 43, 44, 45, 46, 48, 50, 53, 54, 59, 60, 61, 65, 66, 68, 71, 73, 75, 76, 78, 79, 81, 83, 85, 86, 88, 94, 95, 96, 99, 100, 101, 104, 108, 109, 110, 111, 114, 115, 118, 121, 125, 128

Offset: 1

Vincenzo Librandi, Dec 21 2008

One more than the associated term in A024898. - R. J. Mathar, Jan 05 2011

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

Cf. A153245, A153135. - Vladimir Joseph Stephan Orlovsky, Dec 23 2008

[n: n in [1..150] | IsPrime(6*n - 7)]; // Vincenzo Librandi, Sep 24 2012

is(n)=isprime(6*n-7) \\ Charles R Greathouse IV, Feb 20 2017

Corrected and extended by Vladimir Joseph Stephan Orlovsky, Dec 23 2008

A250205 Riesel problem in base 6: Least k > 0 such that n*6^k-1 is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 1, 1, 2, 2, 0, 4, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 3, 0, 1, 1, 6, 2, 0, 5, 1, 1, 1, 0, 6, 2, 1, 1, 0, 1, 2, 10, 1, 0, 1, 3, 1, 1, 0, 1, 1, 2, 1, 0, 1, 8, 1, 1, 0, 1, 2, 2, 4, 0, 49, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 6, 2, 0, 1, 1, 1, 1, 0, 5, 1, 1, 2, 0, 1, 10, 2, 1

Offset: 1

Eric Chen, Mar 13 2015

nonn

a(5j+1) = 0 except for a(1), since (5j+1)*6^k-1 is always divisible by 5, but there are infinitely many numbers not in the form 5j+1 such that a(n) = 0.
a(n) = 0 for n == 84687 mod 10124569, because then n*6^k-1 is always divisible by at least one of 7, 13, 31, 37, 97. - Robert Israel, Mar 17 2015
Conjecture: if n is not in the form 5j+1 and n < 84687, then a(n) > 0.

Eric Chen, Table of n, a(n) for n = 1..1000
Gary Barnes, Riesel conjectures and proofs

Cf. A040081, A046069, A050412, A108129.

Cf. A250204 (Least k > 0 such that n*6^k+1 is prime).

N:= 1000: # to get a(1) to a(N), using k up to 10000
a[1]:= 1:
for n from 2 to N do
if n mod 5 = 1 then a[n]:= 0
else
    for k from 1 to 10000 do
    if isprime(n*6^k-1) then
       a[n]:= k;
         break
      fi
    od
fi
od:
seq(a[n],n=1..N); # Robert Israel, Mar 17 2015

(* m <= 10000 is sufficient up to n = 1000 *)
a[n_] := For[k = 1, k <= 10000, k++, If[PrimeQ[n*6^k - 1], Return[k]]] /. Null -> 0; Table[a[n], {n, 1, 120}]

a(n) = if(n%5==1 && n>1, 0, for(k = 1, 10000, if(ispseudoprime(n*6^k-1), return(k))))

a(A024898(n)) = 1. - Michel Marcus, Mar 16 2015

A307561 Numbers k such that both 6k - 1 and 6k + 5 are prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 14, 17, 18, 22, 28, 29, 32, 38, 39, 42, 43, 44, 52, 58, 59, 64, 74, 77, 84, 93, 94, 98, 99, 107, 108, 109, 113, 137, 143, 147, 157, 158, 162, 163, 169, 182, 183, 184, 197, 198, 203, 204, 213, 214, 217, 227, 228, 238, 239, 247, 248, 249, 259, 267, 268, 269, 312, 317, 318, 329, 333, 344

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 146 terms below 10^3, 831 terms below 10^4, 5345 terms below 10^5, 37788 terms below 10^6 and 280140 terms below 10^7.
Prime pairs differing by 6 are called "sexy" primes. Other prime pairs with difference 6 are of the form 6n + 1 and 6n + 7.
Numbers in this sequence are those which are not 6cd + c - d - 1, 6cd + c - d, 6cd - c + d - 1 or 6cd - c + d, that is, they are not (6c - 1)d + c - 1, (6c - 1)d + c, (6c + 1)d - c - 1 or (6c + 1)d - c.

			a(2) = 2, so 6(2) - 1 = 11 and 6(2) + 5 = 17 are both prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sally M. Moite, "Primeless" Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

Primes differing from each other by 6 are A023201, A046117.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A059325.
Cf. also A307562, A307563.

Select[Range[500], PrimeQ[6# - 1] && PrimeQ[6# + 5] &] (* Alonso del Arte, Apr 14 2019 *)

is(k) = isprime(6*k-1) && isprime(6*k+5); \\ Jinyuan Wang, Apr 20 2019

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 15, 17, 22, 25, 29, 32, 39, 44, 45, 60, 65, 67, 72, 75, 80, 82, 94, 95, 99, 100, 109, 114, 117, 120, 124, 127, 137, 152, 155, 164, 169, 172, 177, 185, 194, 199, 204, 205, 214, 215, 220, 229, 240, 242, 247, 254, 260, 262, 267, 269, 270, 289, 304, 312, 330, 334, 347, 355, 359, 369, 374, 379, 389

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 140 such numbers between 1 and 1000.
These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

			a(4) = 5, so 6(5) - 1 = 29 and 6(5) + 7 = 37 are both prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

The primes are A023202, A092402, A031926.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A153218.
Cf. also A307561, A307562.

select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019

isok(n) = isprime(6*n-1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

Original entry on oeis.org

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

Offset: 1

Eric Desbiaux, Feb 11 2010

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 1 + 3 OR 3* 1 + 2 =  5;
  2* 4 + 3 OR 3* 3 + 2 = 11;
  2* 7 + 3 OR 3* 5 + 2 = 17;
  2*10 + 3 OR 3* 7 + 2 = 23;
  2*13 + 3 OR 3* 9 + 2 = 29;
  2*19 + 3 OR 3*13 + 2 = 41;
  2*22 + 3 OR 3*15 + 2 = 47;
  2*25 + 3 OR 3*17 + 2 = 53;
  2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A007528 Primes of the form 6k-1.
A024898 Positive integers k such that 6k-1 is prime.
1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

Cf. A067076, A024893, A007528, A024898, A059325.

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Data corrected and extended by Amiram Eldar, Jul 30 2024

A223881 Denominators in the expression m!/(prime(m-1)+1) for m > 1 such that this expression is not an integer.

Original entry on oeis.org

3, 2, 19, 31, 37, 79, 41, 97, 53, 139, 71, 157, 83, 199, 211, 229, 131, 271, 137, 307, 331, 337, 173, 367, 379, 197, 439, 227, 499, 263, 547, 281, 577, 293, 197, 199, 601, 607, 619, 661, 227, 229, 691, 239, 727, 383, 269, 811, 829, 283, 431, 877, 467, 937, 313

Offset: 1

Alexander R. Povolotsky, May 01 2013

nonn

It appears that all terms are primes.
From Alexander R. Povolotsky, Apr 26 2025: (Start)
The scatter plot reveals four distinct, well-separated, monotonically increasing curves. It became possible to extract the integers (all conjectured to be primes) corresponding to each of the four subsets.
Additionally, the approximation formulas for each of the four subsets were derived.
These four approximation formulas, given in the exponential form y=C_k*x^m were found to have a common slope: m=1.197311990 while their displacement coefficients are: C_1≈6.86845, C_2≈3.42058, C_3≈2.28335, C_4≈1.70460.
Notably, these displacement coefficients values exhibit a clear pattern: C_2≈C_1/2, C_3≈C_1/3, C_4≈C_1/4. (For instance, 3.42058≈6.86845/2, and so on.)
Above approximations were derived using general separation and approximation methods and do not specifically account for the fact that these values correspond to the prime numbers.
It appears that all primes in the groups 4, 2 and 1 are generated by the 6*k+1 formula, and so primes in the above groups constitute three subsets of A002476 terms, while all primes in the group 3 are generated by the 2*k+1 formula, and so primes in that group constitute a subset of the terms presented in A000040.
Also it appears that:
1. The first group constitutes a sequence, such that for n>=1, a(n) = A005382(n+6).
2. The third group constitutes a sequence, such that for n>1, a(n) = A158015(n+20).
3. The fourth group constitutes a sequence, such that for n>=1, a(n) = A158016(n+32).
The text files containing the primes, corresponding to the above discussed four groups, where primes are indexed against their position in the complete primes listing (see OEIS's A000040), are viewable and downloadable at the below links section. (End)

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Alexander R. Povolotsky, Mathematics StackExchange, Conjecture regarding this sequence, 2025.
Alexander R. Povolotsky, Reindexed group1 primes, 2025.
Alexander R. Povolotsky, Reindexed group2 primes, 2025.
Alexander R. Povolotsky, Reindexed group3 primes, 2025.
Alexander R. Povolotsky, Reindexed group4 primes, 2025.

Cf. A007528, A024898, A002476, A091178.

Denominator[Select[Table[m!/(Prime[m - 1] + 1), {m, 2, 300}], ! IntegerQ[#] &]] (* T. D. Noe, May 03 2013 *)

m=M=1;forprime(p=2,1e5,M*=m++;t=denominator(M/(p+1)); if(t>1, print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

Original entry on oeis.org

1, 4, 5, 14, 15, 29, 39, 40, 49, 70, 110, 159, 169, 204, 235, 260, 264, 315, 334, 355, 390, 425, 449, 490, 560, 565, 599, 634, 725, 729, 735, 820, 824, 889, 1019, 1029, 1349, 1379, 1419, 1510, 1580, 1590, 1694, 1719, 1765, 1925, 1930, 1950, 1985, 2044, 2150

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If k is in the sequence then (6k-1)(12k-1)(18k-1) = 36k * (36k^2 - 11k + 1) - 1 is a Lucas-Carmichael number (A006972).

Analogous to A046025 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers).

			1 is in the sequence since 6*1 - 1 = 5, 12*1 - 1 = 11 and 18*1 - 1 = 17 are all primes, and 5*11*17 = 935 is a Lucas-Carmichael number.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A046025, A067256, A087788, A216925.

Intersection of A024898, A138620 and A138918.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 12 m - 1, 18 m - 1}, PrimeQ ], AppendTo[seq, m] ], {m, 1, 10^5} ]; seq

isok(n) = isprime(6*n-1) && isprime(12*n-1) && isprime(18*n-1); \\ Michel Marcus, Aug 11 2017

6*a(n) - 1 = A067256(n+1).

A330409 Semiprimes of the form p(6p - 1).

Original entry on oeis.org

22, 51, 145, 287, 1717, 2147, 3151, 5017, 11051, 13207, 16801, 20827, 26867, 63551, 68587, 71177, 76501, 96647, 112477, 147737, 159251, 232657, 237407, 308947, 314417, 342487, 433897, 480251, 587501, 602617, 722107, 772927, 834401, 861467, 879751, 907537, 945257, 1155887, 1177051

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

			A158015(1) = 2 is the smallest prime p such that 6p - 1 = 12 - 1 = 11 is also prime, whence a(1) = A049452(2) = 2*(6*2 - 1) = 22.
prime(5) = 11 is the smallest prime not in A024898 or A158015, because 6p - 1 is not a prime, therefore A049452(11) = 11*(6*11 - 1) is not in the sequence, and idem for A049452(13).
But prime(7) = 17 is in A024898 and A158015, so a(5) = A024898(A158015(5)) = A024898(17) = 17*(6*17 - 1).

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

Cf. A024898 (6n-1 is prime), A158015 (primes), A049452 = {n(6n-1)}.

Complement of A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)) in A245365 (primes of the form n(3n-1)/2).

Select[Table[p(6p-1),{p,500}],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 27 2022 *)

[p*(6*p-1) | p<-primes(99), isprime(6*p-1)]

a(n) = A049452(A158015(n)) = p(6p - 1) with p = A158015(n).

cp's OEIS Frontend

A167055 Numbers k such that 12*k + 5 is prime.

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A167057 Numbers k such that 12*k + 11 is prime.

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A153134 Numbers k such that 6k - 7 is prime.

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A250205 Riesel problem in base 6: Least k > 0 such that n*6^k-1 is prime, or 0 if no such k exists.

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A307561 Numbers k such that both 6*k - 1 and 6*k + 5 are prime.

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PARI

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

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Maple

PARI

A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

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A307561 Numbers k such that both 6k - 1 and 6k + 5 are prime.

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.