A060212 -id:A060212 - cp's OEIS frontend

A060213 Lesser of twin primes whose average is 6 times a prime.

11, 17, 29, 41, 101, 137, 281, 617, 641, 821, 1697, 1877, 2081, 2237, 2381, 2657, 2801, 3461, 3557, 3917, 4637, 4721, 5441, 6197, 6701, 8537, 8597, 9677, 10937, 12161, 12377, 12821, 12917, 13217, 13721, 13757, 13997, 14081, 16061, 17417

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Lowest factor-density among all positive consecutive integer triples; for p > 41, last digit of p can be only 1 or 7 (see Alexandrov link, p. 15). - Lubomir Alexandrov, Nov 25 2001

			102197 is here because 102198 = 17033*6 and 17033 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998, see p. 15.

Cf. A002822, A058383, A059960, A027856, A001454, A001359, A060212.

map(t -> 6*t-1, select(p -> isprime(p) and isprime(6*p-1) and isprime(6*p+1), [2,seq(i,i=3..10000,2)]));

Transpose[Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]] == 2 && PrimeQ[Mean[#]/6]&]][[1]] (* Harvey P. Dale, May 04 2014 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 6) && isprime((n+1)/6); \\ Michel Marcus, Dec 14 2013

a(n) = 6 * A060212(n) - 1. - Sean A. Irvine, Oct 31 2022

Offset changed to 1 by Michel Marcus, Dec 14 2013

A294731 Smallest average of a twin prime pair divisible by the n-th prime, i.e. A090530(n), divided by 6*prime(n).

Original entry on oeis.org

1, 1, 3, 4, 1, 2, 1, 2, 7, 9, 5, 4, 1, 20, 3, 43, 4, 3, 14, 22, 9, 8, 19, 7, 1, 1, 8, 4, 24, 5, 1, 2, 2, 13, 4, 6, 5, 9, 22, 3, 15, 6, 11, 3, 7, 5, 20, 5, 6, 7, 3, 3, 9, 14, 10, 2, 35, 2, 1, 10, 25, 17, 1, 35, 5, 4, 1, 18, 15, 12, 25, 1, 2, 5

Offset: 3

Hugo Pfoertner, Nov 09 2017

The sequence starts at n=3, because A090530(1)=4 is not divisible by 6*2 and A090530(2)=6 is not divisible by 6*3.

The positions of ones in the sequence are given by A060212, i.e. a(A000720(A060212(n)))=1 for all n>=3.

			a(5)=3 because 198 is the smallest average of a twin prime pair {197,199} that is divisible by the 5th prime 11: 3 = 198 / (6*11).

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A014574, A060212, A071407, A090530, A182481.

a[n_] := Module[{p = Prime[n], k = 1}, While[! PrimeQ[6*k*p - 1] || ! PrimeQ[6*k*p + 1], k++]; k]; Array[a, 100, 3] (* Amiram Eldar, Aug 25 2025 *)

a(n) = A090530(n) / ( 6 * prime(n) ) for n >= 3.

a(n) = A071407(n) / 6. - Amiram Eldar, Aug 25 2025

A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 37, 43, 47, 53, 67, 73, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 223, 233, 257, 263, 277, 283, 293, 313, 337, 347, 353, 373, 397, 433, 443, 467, 487, 523, 547, 563, 577, 593, 607, 613, 617, 643, 647, 653, 733, 743, 757, 773, 787, 797, 887, 907, 937, 947, 953, 977

Offset: 1

Altug Alkan and Thomas Ordowski, Mar 11 2017

nonn

Note that p - q must be <= 12. Also note that there can be corresponding prime pairs (q, p) more than one way, i.e., (7, 13), (13, 17), (29, 31): (13^2 - 7^2)/24 = (17^2 - 13^2)/24 = (31^2 - 29^2)/24 = 5.
There are no terms of A045468 > 11.
Union of {2}, A006489, A060212, A092110, and A125272. - Robert Israel, Mar 13 2017

			3 is a term since (11^2 - 7^2)/24 = 3 and 3, 7, 11 are prime numbers.

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

Cf. A006489, A060212, A092110, A124865, A124866, A125272, A283532.

select(r -> isprime(r) and ((isprime(3*r+2) and isprime(3*r-2))
  or (isprime(6*r+1) and isprime(6*r-1))
  or (isprime(2*r+3) and isprime(2*r-3))
or (isprime(r+6) and isprime(r-6))), [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 13 2017

ok[n_] := PrimeQ[n] && Block[{p, q, s = Reduce[p^2-q^2 == 24 n && p>3 && q>3, {p, q}, Integers]}, If[s === {}, False, Or @@ And @@@ PrimeQ[{p, q} /. List@ ToRules@s]]]; Select[Range@1000, ok] (* Giovanni Resta, Mar 11 2017 *)

isA124865(n) = if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4, d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0)
lista(nn) = forprime(p=2, nn, if(isA124865(24*p), print1(p", ")))

For n > 5, a(n) == {3,7} mod 10.

A089151 Primes p such that 6p - 7 and 6p - 5 are twin primes.

Original entry on oeis.org

2, 3, 11, 13, 19, 31, 41, 53, 59, 71, 73, 101, 139, 173, 193, 239, 269, 271, 313, 349, 353, 379, 433, 449, 521, 563, 613, 643, 683, 823, 829, 881, 941, 1051, 1061, 1093, 1223, 1249, 1259, 1373, 1399, 1439, 1471, 1571, 1621, 1669, 1723, 1811, 1861, 1951, 1973

Offset: 1

Pierre CAMI, Dec 06 2003

			6*19 - 7 = 107, 6*19 - 5 = 109; 107 and 109 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001097, A001359, A006512, A060212.

[p: p in PrimesUpTo(1000000)| IsPrime(6*p-5) and IsPrime(6*p-7)] // Vincenzo Librandi, Nov 16 2010

Select[Range[2, 2000], And @@ PrimeQ[{#, 6#-5, 6#-7}] &] (* Amiram Eldar, Jan 15 2020 *)
Select[Prime[Range[300]],AllTrue[6#-{5,7},PrimeQ]&] (* Harvey P. Dale, Nov 18 2022 *)

A182521 Numbers n such that A182481(n)=1 and there is not a representation n=d_1*d_2 with d_2>1, such that A182481(d_1)=d_2.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 17, 23, 25, 45, 47, 77, 87, 95, 103, 107, 137, 143, 175, 215, 247, 283, 287, 313, 347, 355, 373, 385, 397, 425, 443, 455, 467, 565, 577, 593, 637, 653, 667, 703, 737, 773, 775, 787, 850, 907, 913, 917, 943, 975, 1033, 1075, 1117, 1127, 1130

Offset: 1

Vladimir Shevelev, May 03 2012

nonn

Or the numbers n such that 6*n-1 is lesser of twin primes which occurs in A182482 only once.

All terms of A060212 are in the sequence.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A182481, A182482, A182483, A182513, A060212.

Numbers n for which A182483(A182513(n))/n = A182481(n) = 1.

Insert 1 and more terms from Ray Chandler, Sep 18 2019

A126655 Numbers n such that 6p(n)-1 and 6p(n)+1 are twin primes and 6p(n+1)-1 and 6p(n+1)+1 are also twin primes with p(n) = n-th prime.

Original entry on oeis.org

1, 2, 3, 27, 137, 340, 479, 882, 1415, 1883, 3442, 3798, 4284, 5827, 7559, 8783, 9453, 10355, 10731, 11388, 12565, 13613, 16477, 17007, 18402, 18665, 19450, 19633, 22306, 24971, 25083, 29108, 29861, 30748, 31694, 32622, 33097, 36743, 37141

Offset: 1

Pierre CAMI, Feb 09 2007

nonn

			6*2-1=11 6*2+1=13 11 13 twin primes as 17 and 19 so 1 is first term of the sequence
6*3-1=17 6*3+1=19 17 19 twin primes as 29 and 31 so 2 is second term of the sequence
6*5-1=29 6*5+1=31 29 and 31 twin primes 5=3rd prime
6*7-1=41 6*7+1=43 41 and 43 twin primes 7=4th prime so 3 is the 3rd term of the sequence

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

Cf. A060212.

Select[Range[39000], PrimeQ[6*Prime[ # ] - 1] && PrimeQ[6*Prime[ # ] + 1] && PrimeQ[6*Prime[ # + 1] - 1] && PrimeQ[6*Prime[ # + 1] + 1] &] (* Ray Chandler, Feb 11 2007 *)
Select[Range[40000],AllTrue[Flatten[{6*Prime[#]+{1,-1},6*Prime[#+1]+{1,-1}}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 27 2015 *)

Corrected and extended by Ray Chandler, Feb 11 2007

A164570 Primes p such that 8p-3 and 8p+3 are also prime numbers.

Original entry on oeis.org

2, 5, 7, 13, 47, 103, 107, 127, 163, 233, 293, 337, 383, 433, 443, 467, 503, 673, 677, 733, 797, 877, 1087, 1093, 1153, 1217, 1223, 1307, 1637, 1933, 2053, 2087, 2137, 2423, 2477, 2543, 2633, 2687, 2857, 2917, 3163, 3373, 3407, 3467, 3767, 3793, 3877

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Subsequence of A023229. [R. J. Mathar, Aug 26 2009]

Primes of the form A087695(k)/8. [R. J. Mathar, Aug 26 2009]

			For p=2, 8*2-3=13 and 8*2+3=19 are prime numbers, which adds p=2 to the sequence
For p=5, 8*5-3=37 and 8*5+3=43 are prime numbers, which adds p=5 to the sequence.

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

Cf. A023212, A023217, A023224, A023230, A023239, A060212, A106015, A124098, A125272, A127430, A153768, A164566, A164567, A164568, A164569.

[p: p in PrimesUpTo(3000) | IsPrime(8*p-3) and IsPrime(8*p+3)]; // Vincenzo Librandi, Apr 09 2013

lst={};Do[p=Prime[n];If[PrimeQ[8*p-3]&&PrimeQ[8*p+3],AppendTo[lst,p]], {n,7!}];lst
Select[Prime[Range[1000]], And@@PrimeQ/@{8 # + 3, 8 # - 3}&] (* Vincenzo Librandi, Apr 09 2013 *)
Select[Prime[Range[1000]],AllTrue[8#+{3,-3},PrimeQ]&] (* Harvey P. Dale, May 05 2023 *)

Comments turned into examples by R. J. Mathar, Aug 26 2009

A171179 Numbers that are divisible by exactly 3 primes (counted with multiplicity) and sandwiched between primes.

Original entry on oeis.org

12, 18, 30, 42, 102, 138, 282, 618, 642, 822, 1698, 1878, 2082, 2238, 2382, 2658, 2802, 3462, 3558, 3918, 4638, 4722, 5442, 6198, 6702, 8538, 8598, 9678, 10938, 12162, 12378, 12822, 12918, 13218, 13722, 13758, 13998, 14082, 16062, 17418, 19542

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 04 2009

nonn

There are 106 of these numbers below 10^5.

			12 is a term: 12 = 2*2*3, and 11 and 13 primes.

Cf. A014612.

Equals 6*A060212. - Zak Seidov and Esko Ranta, Dec 06 2009

Select[Range[9! ],Plus@@Last/@FactorInteger[ # ]==3&&PrimeQ[ #-1]&&PrimeQ[ #+1]&]

A283809 Squarefree numbers k such that 6k - 1 and 6k + 1 are twin primes.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 17, 23, 30, 33, 38, 47, 58, 70, 77, 87, 95, 103, 107, 110, 137, 138, 143, 170, 177, 182, 205, 213, 215, 217, 238, 247, 278, 283, 287, 298, 313, 322, 347, 355, 357, 373, 385, 390, 397, 443, 455, 465, 467, 542, 543, 555, 562, 565, 577, 590, 593, 597, 642, 653, 655, 667, 670, 682

Offset: 1

Juri-Stepan Gerasimov, Mar 17 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002822 and of A005117. Supersequence of A060212.

[n: n in [1..700] | IsSquarefree(n) and IsPrime(6*n-1) and IsPrime(6*n+1)];

Select[Range[100], SquareFreeQ[#] && PrimeQ[6# - 1] && PrimeQ[6# + 1] &] (* Indranil Ghosh, Mar 17 2017 *)

is(n)=isprime(6*n-1) && isprime(6*n+1) && issquarefree(n) \\ Charles R Greathouse IV, Mar 17 2017

A283957 Primes p such that 6p - 1 and 6p + 1 are twin primes and ((6p-1)^2 + (6p+1)^2) / 10 is prime.

Original entry on oeis.org

2, 7, 17, 467, 1033, 2287, 2333, 3413, 7523, 10357, 14723, 15073, 17467, 18077, 19423, 19583, 20177, 24337, 26113, 26357, 26987, 27437, 28627, 29327, 32077, 32323, 33637, 42787, 45127, 46183, 46273, 46457, 53093, 54443, 55333, 57493, 64927, 73363, 75133, 76213, 76493, 76907, 81883, 82633, 86587

Offset: 1

Thomas Ordowski and Altug Alkan, Mar 18 2017

nonn

Only for prime p = 5 there are twin primes 6*5-1 = 29 and 6*5+1 = 31 such that 10 not divides (29^2 + 31^2) = 1802.

			7 is a term because 7, 6*7-1 = 41, 6*7+1 = 43, and (41^2 + 43^2)/10 = 353 are prime numbers.

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

Subsequence of A060212.

Select[Prime@ Range[10^4], Times @@ Boole@ Map[PrimeQ, 6 # + {-1, 1}] == 1 && PrimeQ[((6 # - 1)^2 + (6 # + 1)^2)/10] &] (* Michael De Vlieger, Mar 20 2017 *)
Select[Prime[Range[8500]],AllTrue[{6#-1,6#+1,((6#-1)^2+(6#+1)^2)/10}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 09 2018 *)

a(n) == +-2 (mod 5).

cp's OEIS Frontend

A060213 Lesser of twin primes whose average is 6 times a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A294731 Smallest average of a twin prime pair divisible by the n-th prime, i.e. A090530(n), divided by 6*prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A283562 Primes of the form (p^2 - q^2) / 24 with primes p > q > 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A089151 Primes p such that 6*p - 7 and 6*p - 5 are twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A182521 Numbers n such that A182481(n)=1 and there is not a representation n=d_1*d_2 with d_2>1, such that A182481(d_1)=d_2.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A126655 Numbers n such that 6*p(n)-1 and 6*p(n)+1 are twin primes and 6*p(n+1)-1 and 6*p(n+1)+1 are also twin primes with p(n) = n-th prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A164570 Primes p such that 8*p-3 and 8*p+3 are also prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A089151 Primes p such that 6p - 7 and 6p - 5 are twin primes.

A126655 Numbers n such that 6p(n)-1 and 6p(n)+1 are twin primes and 6p(n+1)-1 and 6p(n+1)+1 are also twin primes with p(n) = n-th prime.

A164570 Primes p such that 8p-3 and 8p+3 are also prime numbers.

A283809 Squarefree numbers k such that 6k - 1 and 6k + 1 are twin primes.