A086801 -id:A086801 - cp's OEIS frontend

A006254 Numbers k such that 2k-1 is prime.

2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157

Offset: 1

Marc LeBrun

a(n) is the inverse of 2 modulo prime(n) for n >= 2. - Jean-François Alcover, May 02 2017
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004
Positions of prime numbers among odd numbers. - Zak Seidov, Mar 26 2013
Also, the integers remaining after removing every third integer following 2, and, recursively, removing every p-th integer following the next remaining entry (where p runs through the primes, beginning with 5). - Pete Klimek, Feb 10 2014
Also, numbers k such that k^2 = m^2 + p, for some integers m and every prime p > 2. Applicable m values are m = k - 1 (giving p = 2k - 1).  Less obvious is: no solution exists if m equals any value in A047845, which is the complement of (A006254 - 1). - Richard R. Forberg, Apr 26 2014
If you define a different type of multiplication (*) where x (*) y = x * y + (x - 1) * (y - 1), (which has the commutative property) then this is the set of primes that follows. - Jason Atwood, Jun 16 2019

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A067076, A086801.
Equals A005097 + 1. A130291 is an essentially identical sequence.
Cf. A065091.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: this seq(k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

[n: n in [0..1000] | IsPrime(2*n-1)]; // Vincenzo Librandi, Nov 18 2010

Rest@Prime@Range@70/2 + 1/2 (* Robert G. Wilson v, Jun 16 2006 *)
Select[Range[200], PrimeQ[2#-1]&] (* Harvey P. Dale, Apr 06 2014 *)

a(n)=prime(n+1)\2+1 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime
def A006254(n): return prime(n+1)+1>>1 # Chai Wah Wu, Aug 02 2024

a(n) = (A000040(n+1) + 1)/2 = A067076(n-1) + 2 = A086801(n-1)/2 + 2.
a(n) = (1 + A065091(n))/2. - Omar E. Pol, Nov 10 2007
a(n) = sqrt((A065091^2 + 2*A065091+1)/4). - Eric Desbiaux, Jun 29 2009
a(n) = A111333(n+1). - Jonathan Sondow, Jan 20 2016

More terms from Erich Friedman

More terms from Omar E. Pol, Nov 10 2007

A067076 Numbers k such that 2*k + 3 is a prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 22, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88, 89, 94, 95, 97, 98, 104, 110, 112, 113, 115, 118, 119, 124, 127, 130, 133, 134, 137, 139, 140, 145, 152, 154, 155

Offset: 1

David Williams, Aug 17 2002

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004
n is in the sequence iff none of the numbers (n-3k)/(2k+1), 1 <= k <= (n-1)/5, is positive integer. - Vladimir Shevelev, May 31 2009
Zeta(s) = Sum_{n>=1} 1/n^s = 1/1 - 2^(-s) * Product_{p=prime=(2*A067076)+3} 1/(1 - (2*A067076+3)^(-s)). - Eric Desbiaux, Dec 15 2009
This sequence is a subsequence of A047949. - Jason Kimberley, Aug 30 2012

Harry J. Smith, Table of n, a(n) for n = 1..1000
Mutsumi Suzuki, Vincenzo Librandi's method for sequential primes (Librandi's description in Italian).

Cf. A086801, A047949.
Numbers n such that 2n+k is prime: A005097 (k=1), this seq(k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19). - Jason Kimberley, Sep 07 2012
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+3) ); # G. C. Greubel, May 21 2019

[n: n in [0..200]| IsPrime(2*n+3)]; // Vincenzo Librandi, Feb 23 2012

select(t -> isprime(2*t+3), [$0..1000]); # Robert Israel, Feb 19 2015

(Prime[Range[100]+1]-3)/2  (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008, modified by G. C. Greubel, May 21 2019 *)
Select[Range[0,200],PrimeQ[2#+3]&] (* Harvey P. Dale, Jun 10 2014 *)

[k | k<-[0..99], isprime(2*k+3)] \\ for illustration

A067076(n) = (prime(n+1)-3)/2 \\ M. F. Hasler, Feb 14 2024

[n for n in (0..200) if is_prime(2*n+3) ] # G. C. Greubel, May 21 2019

a(n) = A006254(n) - 2 = A086801(n+1)/2. [Corrected by M. F. Hasler, Feb 14 2024]
a(n) = A089253(n) - 4. - Giovanni Teofilatto, Dec 14 2003
Conjecture: a(n) = A008507(n) + n - 1 = A005097(n) - 1 = A102781(n+1) - 1. - R. J. Mathar, Jul 07 2009
a(n) = A179893(n) - A000040(n). - Odimar Fabeny, Aug 24 2010

Offset changed from 0 to 1 in 2008: some formulas here and elsewhere may need to be corrected.

A098090 Numbers k such that 2k-3 is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157, 158

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 14 2004

Supersequence of A063908.

Left edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A085090, A086801.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), this sequence (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([1..200], k-> IsPrime(2*k-3) ); # G. C. Greubel, May 21 2019

[ n: n in [1..200] | IsPrime(2*n-3) ]; // Vincenzo Librandi, Dec 26 2010

(Prime[Range[2,100]]+3)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[200],PrimeQ[2#-3]&] (* Harvey P. Dale, Mar 05 2022 *)

is(n)=isprime(2*n-3) \\ Charles R Greathouse IV, Feb 17 2017

[n for n in (1..200) if is_prime(2*n-3) ] # G. C. Greubel, May 21 2019

Half of p + 3, where p is a prime greater than 2.

A122845(a(n), 3) = 3; a(n) = A113935(n+1)/2. - Reinhard Zumkeller, Sep 14 2006

A097069 Positive integers n such that 2n - 9 is prime.

Original entry on oeis.org

6, 7, 8, 10, 11, 13, 14, 16, 19, 20, 23, 25, 26, 28, 31, 34, 35, 38, 40, 41, 44, 46, 49, 53, 55, 56, 58, 59, 61, 68, 70, 73, 74, 79, 80, 83, 86, 88, 91, 94, 95, 100, 101, 103, 104, 110, 116, 118, 119, 121, 124, 125, 130, 133, 136, 139, 140, 143, 145, 146, 151, 158, 160

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 15 2004

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

Cf. A000040, A086801.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), this seq(k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

[n: n in [6..160] | IsPrime(2*n-9)];  // Bruno Berselli, Mar 05 2011

(Prime[Range[2,100]]+9)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[4, 200], PrimeQ[2 # - 9] &] (* Vincenzo Librandi, Oct 16 2012 *)

is(n)=isprime(2*n-9) \\ Charles R Greathouse IV, Apr 28 2015

Half of p+9 where p is a prime greater than 2.

A172367 Numbers k > 0 such that k+4 is a prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 25, 27, 33, 37, 39, 43, 49, 55, 57, 63, 67, 69, 75, 79, 85, 93, 97, 99, 103, 105, 109, 123, 127, 133, 135, 145, 147, 153, 159, 163, 169, 175, 177, 187, 189, 193, 195, 207, 219, 223, 225, 229, 235, 237, 247, 253, 259, 265, 267, 273, 277, 279

Offset: 1

Juri-Stepan Gerasimov, Feb 01 2010

The subsequence of primes A023200 consists of the smallest primes p of cousin prime pairs (p, p+4), while the subsequence of nonprimes is A164384. - Bernard Schott, Oct 19 2021

			a(1) = 5 - 4 = 1, a(2) = 7 - 4 = 3.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Union of A023200 and A164384.

Cf. A000040, A006093, A040976, A086801.

Table[Prime[n]-4,{n,3,53}] (* Charles R Greathouse IV, Mar 12 2012 *)

a(n)=prime(n+2)-4 \\ Charles R Greathouse IV, Mar 12 2012

a(n) = prime(n+2) - 4.

A175222 a(n) = prime(n) + 5.

Original entry on oeis.org

7, 8, 10, 12, 16, 18, 22, 24, 28, 34, 36, 42, 46, 48, 52, 58, 64, 66, 72, 76, 78, 84, 88, 94, 102, 106, 108, 112, 114, 118, 132, 136, 142, 144, 154, 156, 162, 168, 172, 178, 184, 186, 196, 198, 202, 204, 216, 228, 232, 234, 238, 244, 246, 256, 262, 268, 274, 276

Offset: 1

Jaroslav Krizek, Mar 06 2010

a(n) = A000040(n) + 5 = A008864(n) + 4 = A052147(n) + 3 = A113395(n) + 2 = A175221 (n) + 1 = A139049(n) - 1 = A175223(n) - 2 = A175224(n) - 3 = A140353(n) - 4 = A175225(n) - 5.

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

Cf. A000040, A008864, A052147, A086801, A113395, A113935, A139049, A140353, A175221, A175223, A175224, A175225.

Filtered([1..300], k-> IsPrime(k) ) + 5; # G. C. Greubel, May 20 2019

[p+5: p in PrimesUpTo(500)]; // Vincenzo Librandi, Dec 04 2010

Prime[Range[120]]+5 (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)

{a(n) = prime(n) + 5}; \\ G. C. Greubel, May 20 2019

[nth_prime(n) + 5 for n in (1..100)] # G. C. Greubel, May 20 2019

More terms from Vincenzo Librandi, Mar 14 2010

A244798 Number of moduli m such that (prime(n) mod m) = 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 3, 4, 2, 4, 2, 2, 6, 4, 4, 6, 2, 5, 4, 6, 4, 8, 2, 2, 4, 7, 6, 2, 6, 4, 6, 2, 6, 2, 4, 6, 10, 4, 6, 8, 2, 4, 6, 2, 7, 8, 10, 10, 2, 6, 4, 6, 6, 2, 10, 6, 4, 2, 2, 14, 6, 8, 10, 6, 2, 6, 2, 6, 2, 10, 4, 10, 6, 6, 10, 2, 2, 2, 6, 10, 6, 4

Offset: 1

Clark Kimberling, Jul 06 2014

Except for the initial 0, 0, 0, this is column 3 of the array at A244740.

			prime(5) = 11 = (2 mod m) for m = 3, 9 so that a(5) = 2.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A244740, A244796, A244797.

Cf. A070824, A086801.

z = 300; f[n_, m_] := If[Mod[Prime[n], m] == 3, 1, 0];
t = Table[f[n, m], {n, 1, z}, {m, 1, Prime[n]}];
Table[Count[t[[k]], 1], {k, 1, z}] (* A244798 *)

a(n) = A070824(A086801(n)), for n > 2. - Ridouane Oudra, Mar 17 2024

A190136 Largest prime factor of n(n+1)(n+2)*(n+3).

Original entry on oeis.org

3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 13, 13, 29, 29, 31, 31, 31, 31, 17, 17, 37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 17, 17, 53, 53, 53, 53, 19, 29, 59, 59, 61, 61, 61, 61, 31, 13, 67, 67, 67

Offset: 1

Reinhard Zumkeller, May 07 2011

nonn

a(n) > 11 for n > 9;
a(A086801(n)) = A000040(n) for n > 2.
It follows from Størmer's theorem that lim inf a(n) = infinity, and in fact a(n) >> log log n. - Charles R Greathouse IV, Feb 19 2013

			Numbers m <= 10^6 such that a(m) = p:
p=13: 10, 11, 12, 13, 24, 25, 63;
p=17: 14, 15, 32, 33, 48, 49;
p=19: 16, 17, 18, 19, 54, 75, 168;
p=23: 20, 21, 22, 23, 207, 322;
p=29: 26, 27, 55, 114;
p=31: 28, 29, 30, 31, 62, 90, 152, 153, 340, 493, 1518;
p=37: 34, 35, 36, 37, 74, 184, 405;
p=41: 38, 39, 123, 245, 285, 286, 287, 492, 1023, 1517, 1680;
p=43: 40, 41, 42, 43, 84, 85, 169, 258, 341, 342, 558, 1330, 1331, 2106, 5289, 10878;
p=47: 44, 45, 46, 47, 91, 92, 93, 185, 186, 187, 374, 375, 702, 986, 987, 17575;
p=53: 50, 51, 52, 53, 159, 368, 369, 527, 845, 899, 900, 1375;
p=59: 56, 57, 115, 116, 117, 118, 174, 294, 528, 529, 530, 648, 943, 1885, 6783;
p=61: 58, 59, 60, 61, 119, 120, 121, 122, 182, 183, 242, 243, 244, 549, 608, 609, 1034, 1218, 1219, 1767, 1768, 2013, 2254, 2622;
p=67: 64, 65, 66, 67, 132, 133, 735, 1271, 1272, 1273, 2208, 2277, 3885, 4958, 5828, 5829;
p=71: 68, 69, 140, 141, 142, 284, 423, 424, 494, 636, 637, 779, 780, 781, 3477, 3478, 3549, 3550, 4899;
p=73: 70, 71, 72, 73, 143, 144, 145, 219, 363, 510, 728, 729, 803, 1022, 1239, 1679, 2772, 70224;
p=79: 76, 77, 78, 79, 158, 234, 235, 472, 473, 474, 550, 867, 868, 1024, 1104, 1419, 2209, 2448, 2923, 3476, 3869, 4898, 5290, 7502, 46136, 70150;
p=83: 80, 81, 82, 83, 246, 247, 413, 495, 663, 664, 1078, 1159, 1824, 2736, 3483, 4232, 4896, 4897, 7137, 8214, 12614, 36517, 97524;
p=89: 86, 87, 88, 89, 175, 264, 265, 354, 531, 710, 711, 712, 798, 1245, 1332, 2847, 4895, 5073, 6318, 18423, 28302, 29279;
p=97: 94, 95, 96, 97, 288, 289, 483, 580, 581, 582, 774, 873, 1064, 1065, 1455, 2132, 2133, 3007, 3975, 4556, 4557, 6496, 6497, 6887, 7564, 7565, 7566, 13869, 17457.

Paulo Ribenboim, Galimatias Arithmeticae (Chap 11), in 'My Numbers, My Friends', Springer-Verlag 2000 NY, page 345.
J. J. Sylvester, "On arithmetical series", Messenger of Mathematics 21 (1892), pp. 1-19 and 87-120.
M. Faulkner, "On a theorem of Sylvester and Schur", J. London Math. Soc. 41:1 (1966), pp. 107-110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A074399, A093074, A193945.

a190136 n = maximum $ map a006530 [n..n+3]

Table[FactorInteger[Times@@(n+Range[0,3])][[-1,1]],{n,70}] (* Harvey P. Dale, Mar 19 2018 *)

gpf(n)=vecmax(factor(n)[,1])
a(n)=my(p=precprime(n+3));if(pCharles R Greathouse IV, Feb 19 2013

a(n) = max{gpf(n), gpf(n+1), gpf(n+2), gpf(n+3)} = gpf(A052762(n+3)) with gpf = A006530, greatest prime factor.

a(n) > 47 for n > 17575. - Charles R Greathouse IV, Feb 19 2013

A144842 Numbers k such that the three numbers k+3, k-3 and k+5 are all prime.

Original entry on oeis.org

8, 14, 26, 56, 104, 134, 176, 194, 236, 266, 566, 596, 656, 824, 1016, 1226, 1286, 1484, 1604, 1616, 1874, 2084, 2336, 2546, 2966, 3254, 3326, 3464, 3536, 3764, 3914, 3926, 4016, 4214, 4256, 4646, 4796, 5006, 5276, 5474, 5654, 5846, 5864, 6266, 6356, 6566

Offset: 1

Giovanni Teofilatto, Sep 22 2008

Subset of A087695. - R. J. Mathar, Sep 24 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A046138, A087695, A086801, A144840.

Select[Range[7000], And @@ PrimeQ[# + {-3, 3, 5}] &] (* Amiram Eldar, Apr 14 2022 *)

from sympy import isprime
def ok(n): return n > 4 and isprime(n-3) and isprime(n+3) and isprime(n+5)
print(list(filter(ok, range(6567)))) # Michael S. Branicky, Aug 14 2021

a(n) = A046138(n) + 3. - R. J. Mathar, Sep 24 2008

Definition edited and extended by R. J. Mathar, Sep 24 2008

A154115 Numbers n such that n + 3 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 16, 20, 26, 28, 34, 38, 40, 44, 50, 56, 58, 64, 68, 70, 76, 80, 86, 94, 98, 100, 104, 106, 110, 124, 128, 134, 136, 146, 148, 154, 160, 164, 170, 176, 178, 188, 190, 194, 196, 208, 220, 224, 226, 230, 236, 238, 248, 254, 260, 266, 268, 274, 278

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

			a(2) = 2 since (2 + 2)^2 - (2 + 1)^2 - 2 = 5.

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097, A154111, A154112, A154113.

Cf. A067076 (a(n-1)/2).

[n: n in [0..500] | IsPrime((n+2)^2-(n+1)^2-n)]; // Vincenzo Librandi, Nov 26 2010

A154115 := proc(n) ithprime(n+1)-3 ; end proc: # R. J. Mathar, May 09 2010

a[n_]:=(n+2)^2-(n+1)^2-n;lst={};Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,6!}];lst
Select[Range[0,300],PrimeQ[(#+2)^2-(#+1)^2-#]&] (* Harvey P. Dale, Nov 06 2013 *)
Prime[Range[2,100]]-3 (* Harvey P. Dale, Jul 15 2017 *)

is(n)=isprime(n+3) \\ Charles R Greathouse IV, Sep 02 2016

a(n) = A086801(n+1). - R. J. Mathar, May 09 2010

New name based on a comment by Franklin T. Adams-Watters, Jan 30 2009

cp's OEIS Frontend

A006254 Numbers k such that 2k-1 is prime.

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A067076 Numbers k such that 2*k + 3 is a prime.

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A098090 Numbers k such that 2k-3 is prime.

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A097069 Positive integers n such that 2n - 9 is prime.

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A172367 Numbers k > 0 such that k+4 is a prime.

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A175222 a(n) = prime(n) + 5.

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A190136 Largest prime factor of n(n+1)(n+2)*(n+3).