A060229 -id:A060229 - cp's OEIS frontend

A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)q(n)-1 and a(n)q(n)+1 are a pair of twin primes.

2, 1, 1, 2, 1, 6, 8, 11, 4, 16, 22, 4, 74, 24, 37, 28, 14, 11, 242, 11, 91, 20, 83, 91, 35, 80, 48, 47, 226, 2, 12, 203, 30, 38, 356, 54, 266, 108, 305, 227, 173, 1185, 738, 13, 382, 277, 455, 433, 173, 1303, 926, 1162, 164, 298, 69, 121, 702, 1670, 36, 570, 170, 204

Offset: 1

Labos Elemer, Mar 22 2001

nonn

			30030*j-1 or 30030*j+1 are not both primes for j=1,2,3,4,5. But for j=6 {180179,180181} are twin primes. So a(6)=6.

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A001359, A002110, A060229-A060232, A060255, A057706.

smp[n_]:=Module[{k=1},While[!PrimeQ[k*n+1]||!PrimeQ[k*n-1],k++];k]; Table[ smp[n],{n,FoldList[Times,Prime[Range[70]]]}] (* Harvey P. Dale, Oct 27 2016 *)

a(n)=p=vecprod(primes(n));for(k=1,+oo,ispseudoprime(k*p+1)&&ispseudoprime(k*p-1)&&return(k)) \\ Jeppe Stig Nielsen, Nov 09 2024

Corrected and extended by Ray Chandler, Apr 03 2009

A158277 The lesser of twin prime pairs with each prime in a different century.

Original entry on oeis.org

599, 2999, 3299, 4799, 5099, 6299, 8999, 10499, 11699, 21599, 25799, 26699, 29399, 33599, 34499, 36899, 37199, 42899, 44699, 47699, 49199, 56099, 57899, 60899, 63599, 65099, 65699, 70199, 74099, 81899, 83399, 85199, 88799, 92399, 97499, 100799, 101999, 102299

Offset: 1

Ki Punches, Mar 15 2009

nonn

The sequence is conjecturally infinite; note that those ending in 999 straddle millenia: A158861.

Since any prime greater than 3 is congruent to 1 or 5 modulo 6, a(n)+1 is divisible by 300 (see A001359). - Hartmut F. W. Hoft, May 18 2017

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

Cf. A060229, A158861, A001359, A060229.

Select[Range[100,110000,100],AllTrue[#+{1,-1},PrimeQ]&]-1 (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)
a158277[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[100, n, 100]], First[#]-Last[#]==2&]]
a158277[105000] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)

Corrected by Ray Chandler and R. J. Mathar, Apr 03 2009

A282321 Lesser of twin primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 191, 281, 311, 431, 461, 521, 641, 821, 881, 1031, 1061, 1091, 1151, 1301, 1451, 1481, 1721, 1871, 1931, 2081, 2111, 2141, 2381, 2591, 2711, 2801, 3251, 3371, 3461, 3581, 3671, 3821, 3851, 4001, 4091, 4241, 4271, 4421, 4481, 4721, 4931, 5021

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [this sequence and A282322] is A132241.
The union of [{3, 5}, this sequence, A282323 and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subset of A001097, A001359, A132232, A132241 and A132247.

Cf. A006512, A232880, A232881, A232882, A282322, A282323, A060229, A060229, A282326.

[p: p in PrimesUpTo(5000) | IsPrime(p+2) and p mod 30 eq 11 ]; // Vincenzo Librandi, Feb 12 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 11 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==13, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A158861 The lesser of twin prime pairs with each prime in a different millennia.

Original entry on oeis.org

2999, 8999, 101999, 164999, 179999, 230999, 272999, 293999, 326999, 389999, 410999, 419999, 443999, 512999, 524999, 536999, 659999, 662999, 773999, 788999, 794999, 800999, 818999, 890999, 920999, 932999, 989999, 1028999, 1058999, 1136999, 1187999, 1238999

Offset: 1

Ki Punches, Mar 28 2009

nonn

The sequence is conjecturally infinite; note that terms ending 9999 straddle ten millennia.

Cf. A060229, A158277.

Corrected and extended by Ray Chandler and R. J. Mathar, Apr 03 2009

A282323 Lesser of twin primes congruent to 17 (mod 30).

Original entry on oeis.org

17, 107, 137, 197, 227, 347, 617, 827, 857, 1277, 1427, 1487, 1607, 1667, 1697, 1787, 1877, 1997, 2027, 2087, 2237, 2267, 2657, 2687, 3167, 3257, 3467, 3527, 3557, 3767, 3917, 4127, 4157, 4217, 4337, 4517, 4547, 4637, 4787, 4967, 5417, 5477, 5657, 5867, 6197

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [this sequence and A282324] is A132242.
The union of [{3, 5}, A282321, this sequence and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
A181605 without the 7. The proof works along the same lines as the proof in A282322. - R. J. Mathar, Feb 14 2017
Number of terms < 10^k: 0, 0, 1, 9, 64, 414, 2734, 19674, 146953, ... - Muniru A Asiru, Jan 09 2018

			From _Muniru A Asiru_, Jan 25 2018: (Start)
17 is a member because the pair (17, 19) is a twin prime, 17 < 19 and 17 mod 30 = 17.
137 is a member because the pair (137, 139) is a twin prime, 137 < 139 and 137 mod 30 = 17.
197 is a member because the pair (197, 199) is a twin prime, 197 < 199 and 197 mod 30 = 17.
(End)

Muniru A Asiru, Table of n, a(n) for n = 1..3000

Subset of A001097, A001359, A039949, A132242 and A132247.

Cf. A006512, A232880, A232881, A232882, A282321, A282322, A282324, A282326.

P:=Filtered([1..400000], IsPrime);;
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=2),j->j[1] mod 30=17),k->k[1]);; # Muniru A Asiru, Jul 08 2017

[p: p in PrimesUpTo(7000) | IsPrime(p+2) and p mod 30 eq 17 ]; // Vincenzo Librandi, Feb 13 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 17 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

Select[17 + 30 Range[0, 220], PrimeQ[#] && PrimeQ[# + 2] &] (* Robert G. Wilson v, Jan 09 2018 *)
Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&&Mod[#[[1]],30]==17&][[;;,1]] (* or *) Select[Range[17,7000,30],AllTrue[#+{0,2},PrimeQ]&] (* Harvey P. Dale, Mar 02 2024 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==19, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A282324 Greater of twin primes congruent to 19 (mod 30).

Original entry on oeis.org

19, 109, 139, 199, 229, 349, 619, 829, 859, 1279, 1429, 1489, 1609, 1669, 1699, 1789, 1879, 1999, 2029, 2089, 2239, 2269, 2659, 2689, 3169, 3259, 3469, 3529, 3559, 3769, 3919, 4129, 4159, 4219, 4339, 4519, 4549, 4639, 4789, 4969, 5419, 5479, 5659, 5869, 6199

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A282323 and this sequence] is A132242.
The union of [{5, 7}, A282322, this sequence and A282326] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.
Number of terms less than 10^k, k=2,3,4,...: 1, 9, 64, 414, 2734, 19674, 146953, ... - Muniru A Asiru, Feb 09 2018

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Subset of A001097, A006512, A132234, A132242 and A132247.

Cf. A001359, A132235, A132236, A232880, A232881, A232882, A282321, A282322, A282323, A060229, A282326.

Filtered(List([1..220], k -> 30*k-11), n -> IsPrime(n) and IsPrime(n-2));  # Muniru A Asiru, Feb 02 2018

[p: p in PrimesUpTo(7000) | IsPrime(p-2) and p mod 30 eq 19 ]; // Vincenzo Librandi, Feb 13 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)-2) and ithprime(i) mod 30 = 19 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;
# More efficient
select(n -> isprime(n-2) and isprime(n), [seq(30*k+19, k=0..220)]); # Muniru A Asiru, Jan 30 2018

Select[Prime[Range[1000]], PrimeQ[# - 2] && Mod[#, 30] == 19 &] (* Vincenzo Librandi, Feb 13 2017 *)

list(lim)=my(v=List(), p=2); forprime(q=3, lim, if(q-p==2 && q%30==19, listput(v, q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A282326 Greater of twin primes congruent to 1 (mod 30).

Original entry on oeis.org

31, 61, 151, 181, 241, 271, 421, 571, 601, 661, 811, 1021, 1051, 1231, 1291, 1321, 1621, 1951, 2131, 2311, 2341, 2551, 2731, 2791, 2971, 3001, 3121, 3301, 3331, 3361, 3391, 3541, 3931, 4021, 4051, 4231, 4261, 4651, 4801, 5011, 5101, 5281, 5521, 5641, 5851

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [A060229 and this sequence] is A132243.
The union of [{5, 7}, A282322, A282324 and this sequence] is the greater of twin primes sequence A006512.
The union of [{3, 5, 7}, A282321 to A060229 and this sequence] is the twin primes sequence A001097.
Number of terms less than 10^k, k=2,3,4,...: 2, 11, 72, 407, 2697, 19507, 146516, ... - Muniru A Asiru, Mar 05 2018

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Subset of A006512, A181603, A132230, A132243 and A132247.

Cf. A001359, A232880, A232881, A232882, A282321, A282322, A282323, A282324, A060229.

Filtered(List([0..300], k -> 30*k+1), n -> IsPrime(n-2) and IsPrime(n));  # Muniru A Asiru, Mar 05 2018

select(n -> isprime(n-2) and isprime(n), [seq(30*k+1, k=0..300)]); # Muniru A Asiru, Mar 05 2018

1 + Select[30 Range@ 200, AllTrue[# + {-1, 1}, PrimeQ] &] (* Michael De Vlieger, Mar 26 2018 *)

list(lim)=my(v=List(),p=2); forprime(q=3,lim, if(q-p==2 && q%30==1, listput(v,q)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A060230 Smaller of twin primes whose middle term is a multiple of A002110(4)=210.

Original entry on oeis.org

419, 1049, 2309, 2729, 3359, 5879, 6089, 6299, 7349, 7559, 8819, 9239, 10499, 10709, 11549, 11969, 15329, 18059, 21839, 25409, 26249, 26879, 28349, 29399, 30869, 31079, 32969, 33179, 33599, 33809, 34649, 35279, 37589, 40529, 42839

Offset: 1

Labos Elemer, Mar 21 2001

nonn

Number of terms less than 10^k: 0, 0, 0, 1, 12, 80, 542, 3908, 29229, ... - Muniru A Asiru, Jan 29 2018

			For the pair {1049,1051} (1049+1051)/2 = 5*210.

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

A001359, A002110, A060229, A060231.

P:=Filtered([1..10^7], IsPrime);;
P1:=List(Filtered(Filtered(List([1..Length(P)-1], n -> [P[n],P[n+1]]), i -> i[2]-i[1]=2), j -> (j[1]+j[2]) mod 210 = 0), k -> k[1]); # Muniru A Asiru, Jan 29 2018

select(n->isprime(n) and isprime(n+2), [seq(210*k - 1, k=1..10^3)]); # Muniru A Asiru, Jan 29 2018

Select[210*Range[250],And@@PrimeQ[#+{1,-1}]&]-1 (* Harvey P. Dale, Aug 25 2013 *)

is(n)=(n+1)%210==0 && isprime(n+2) && isprime(n) \\ Charles R Greathouse IV, Jan 29 2018

Minor edits by Ray Chandler, Apr 02 2009

A060231 Smaller of twin primes whose middle term is a multiple of A002110(5)=2310.

Original entry on oeis.org

2309, 9239, 11549, 25409, 34649, 43889, 55439, 78539, 92399, 110879, 117809, 133979, 152459, 168629, 180179, 224069, 226379, 230999, 244859, 251789, 267959, 270269, 284129, 297989, 300299, 309539, 314159, 316469, 330329, 376529, 390389

Offset: 1

Labos Elemer, Mar 21 2001

nonn

Number of terms less than 10^k: 0, 0, 0, 0, 2, 9, 66, 422, 3255, ... - Muniru A Asiru, Jan 29 2018

			For the pair {9239,9241} (9239+9241)/2 = 4*2310.

Muniru A Asiru, Table of n, a(n) for n = 1..1000

Cf. A001359, A002110, A060229, A060230.

P:=Filtered([1..10^5], IsPrime);;
P1:=List(Filtered(Filtered(List([1..Length(P)-1], n -> [P[n],P[n+1]]),i -> i[2]-i[1] = 2), j -> (j[1]+j[2]) mod 2310 = 0), k -> k[1]); # Muniru A Asiru, Jan 29 2018

select(n->isprime(n) and isprime(n+2), [seq(2310*k-1, k=1..10^3)]);  # Muniru A Asiru, Jan 29 2018

Select[2310*Range[200],And@@PrimeQ[#+{1,-1}]&]-1 (* Harvey P. Dale, Aug 23 2013 *)

is(n)=(n+1)%2310==0 && isprime(n+2) && isprime(n) \\ Charles R Greathouse IV, Jan 30 2018

Minor edits by Ray Chandler, Apr 02 2009

A181606 Twin primes ending in 9.

Original entry on oeis.org

19, 29, 59, 109, 139, 149, 179, 199, 229, 239, 269, 349, 419, 569, 599, 619, 659, 809, 829, 859, 1019, 1049, 1229, 1279, 1289, 1319, 1429, 1489, 1609, 1619, 1669, 1699, 1789, 1879, 1949, 1999, 2029, 2089, 2129, 2239, 2269, 2309, 2339, 2549, 2659, 2689

Offset: 1

Omar E. Pol, Nov 01 2010

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A001097, A181603, A181604, A181605.

Union of A060229 and A282324.

filter:= proc(n)
  if not isprime(n) then return false fi;
  if n mod 3 = 1 then isprime(n-2) else isprime(n+2) fi
end proc:
select(filter, [seq(i,i=9..10^4,10)]); # Robert Israel, Nov 19 2023

Select[Prime@ Range@ 800, Mod[ #, 10] == 9 && (PrimeQ[ # - 2] || PrimeQ[ # + 2]) &] (* Robert G. Wilson v, Nov 06 2010 *)
Select[Union[Flatten[Select[Partition[Prime[Range[400]],2,1],#[[2]]-#[[1]]==2&]]],Mod[#,10]==9&] (* Harvey P. Dale, May 21 2024 *)

More terms from Robert G. Wilson v, Nov 06 2010

cp's OEIS Frontend

A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)*q(n)-1 and a(n)*q(n)+1 are a pair of twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A158277 The lesser of twin prime pairs with each prime in a different century.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A282321 Lesser of twin primes congruent to 11 (mod 30).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

PARI

A158861 The lesser of twin prime pairs with each prime in a different millennia.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A282323 Lesser of twin primes congruent to 17 (mod 30).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

A282324 Greater of twin primes congruent to 19 (mod 30).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

A282326 Greater of twin primes congruent to 1 (mod 30).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

A060230 Smaller of twin primes whose middle term is a multiple of A002110(4)=210.

A060256 Smallest multiple a(n) of n-th primorial q(n) such that a(n)q(n)-1 and a(n)q(n)+1 are a pair of twin primes.