A181605 -id:A181605 - cp's OEIS frontend

A181603 Twin primes ending in 1.

11, 31, 41, 61, 71, 101, 151, 181, 191, 241, 271, 281, 311, 421, 431, 461, 521, 571, 601, 641, 661, 811, 821, 881, 1021, 1031, 1051, 1061, 1091, 1151, 1231, 1291, 1301, 1321, 1451, 1481, 1621, 1721, 1871, 1931, 1951, 2081, 2111, 2131, 2141, 2311, 2341, 2381

Offset: 1

Omar E. Pol, Nov 01 2010

These are twin primes == 1 (mod 30) or == 11 (mod 30) or == 21 (mod 30). In the first case they cannot be lower twin primes because the upper ones would be == 3 (mod 30) and divisible by 3. In the second case they cannot be upper twin primes because the lower ones would be == 9 (mod 30) and divisible by 3. The last case is excluded because that implies they are divisible by 3. In summary the upper twin primes in here are given by A282326, the lower twin primes in here by A282321. - R. J. Mathar, Feb 14 2017

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A001097, A181604, A181605, A181606.

isA181603 := proc(p)
    if isprime(p) and (isprime(p-2) or isprime(p+2)) then
        if modp(p,10) = 1 then
            true;
        else
            false;
        end if ;
    else
        false;
    end if;
end proc:
for n from 1 to 1000 do
    if isA181603(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 14 2017

Select[Prime@ Range@ 360, Mod[ #, 10] == 1 && (PrimeQ[ # - 2] || PrimeQ[ # + 2]) &] (* Robert G. Wilson v, Nov 06 2010 *)
Select[Flatten[Select[Partition[Prime[Range[400]],2,1],#[[2]]-#[[1]] == 2&]],Mod[#,10]==1&] (* Harvey P. Dale, Oct 24 2021 *)

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

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

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

A181604 Twin primes ending in 3.

Original entry on oeis.org

3, 13, 43, 73, 103, 193, 283, 313, 433, 463, 523, 643, 823, 883, 1033, 1063, 1093, 1153, 1303, 1453, 1483, 1723, 1873, 1933, 2083, 2113, 2143, 2383, 2593, 2713, 2803, 3253, 3373, 3463, 3583, 3673, 3823, 3853, 4003, 4093, 4243, 4273, 4423, 4483, 4723, 4933

Offset: 1

Omar E. Pol, Nov 01 2010

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos

Cf. A001097, A181603, A181605, A181606.

isA181604 := proc(p)
    if isprime(p) and (isprime(p-2) or isprime(p+2)) then
        if modp(p,10) = 3 then
            true;
        else
            false;
        end if ;
    else
        false;
    end if;
end proc: # R. J. Mathar, Feb 14 2017

Select[Prime@Range@700, Mod[ #, 10] == 3 && (PrimeQ[ # - 2] || PrimeQ[ # + 2]) &] (* Robert G. Wilson v, Nov 06 2010 *)

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

A253624 Initial members of prime sextuples (p, p+2, p+12, p+14, p+24, p+26).

Original entry on oeis.org

5, 17, 1277, 4217, 21587, 91127, 103967, 113147, 122027, 236867, 342047, 422087, 524957, 560477, 626597, 754967, 797567, 909317, 997097, 1322147, 1493717, 1698857, 1748027, 1762907, 2144477, 2158577, 2228507, 2398157, 2580647, 2615957

Offset: 1

Karl V. Keller, Jr., Jan 06 2015

nonn

This sequence is primes p for which there exist three twin prime pairs (p, p+2), (p+12, p+14) and (p+24, p+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30n+17). A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
Note that not in all cases (p, p+2, p+12, p+14, p+24, p+26) are consecutive primes; the first p's for which (p, p+2, p+12, p+14, p+24, p+26) are consecutive primes are 4217, 21587, 91127, 103967, 236867, 342047, 422087, 560477, 797567, 909317, 1322147, 1493717, 1748027, 1762907, 2144477, 2158577, 2228507, 2615957 (not in OEIS). - Zak Seidov, May 16 2017

			For p = 17, the numbers 17, 19, 29, 31, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

select(t -> andmap(isprime, [t,t+2,t+12,t+14,t+24,t+26]),
[5, seq(30*k+17,k=0..10^5)]); # Robert Israel, Jan 07 2015

Select[Prime@ Range[2*10^5], Times @@ Boole@ PrimeQ[# + {2, 12, 14, 24, 26}] == 1 &] (* Michael De Vlieger, May 16 2017 *)
Select[Prime[Range[200000]],AllTrue[#+{2,12,14,24,26},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 15 2021 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+12) and isprime(n+14) and isprime(n+24) and isprime(n+26): print(n,end=', ')

from sympy import isprime, primerange
def aupto(limit):
  alst = []
  for p in primerange(2, limit+1):
    if all(map(isprime, [p+2, p+12, p+14, p+24, p+26])): alst.append(p)
  return alst
print(aupto(3*10**6)) # Michael S. Branicky, May 17 2021

A092340 Prime numbers n such that n^2+2n divides (Fibonacci(n^2) + Fibonacci(2n)).

Original entry on oeis.org

7, 17, 107, 137, 197, 227, 347, 617, 827, 857, 1277, 1427, 1487, 1607, 1667, 1697, 1787, 1877, 1997, 2027, 2087, 2237, 2267, 2657, 2687, 2707, 3167, 3257, 3467, 3527, 3557, 3767, 3917, 4127, 4157, 4217, 4337, 4517, 4547, 4637, 4787, 4967, 5417, 5477

Offset: 1

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 18 2004

nonn

First disagrees with A181605 at n=26: this sequence contains 2707, but A181605 doesn't. Is this a supersequence of A181605? - Nathaniel Johnston, Jun 25 2011

See link for proof of this. - Robert Israel, Apr 13 2021

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 182 terms from Robert G. Wilson v)
Robert Israel, Proof that A181605 is a subsequence of A092340

Cf. A000045.

Cf. A181605. - Robert G. Wilson v, Nov 07 2010

filter:= proc(n)
  local M,A;
  uses LinearAlgebra:-Modular;
  if not isprime(n) then return false fi;
  M:= Matrix(2,2,<<0,1>|<1,1>>,datatype=integer);
  A:= MatrixPower(n,M,n^2) + MatrixPower(n,M,2*n);
  if A[1,2] mod n <> 0 then return false fi;
  A:= MatrixPower(n+2,M,n^2) + MatrixPower(n+2,M,2*n);
  A[1,2] mod (n+2) = 0
end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Apr 13 2021

fQ[n_] := Mod[ Fibonacci[n^2] + Fibonacci[2 n], n^2 + 2 n] == 0; Select[ Prime@ Range@ 744, fQ] (* Robert G. Wilson v, Nov 07 2010 *)

forprime (i=1,2000,if(Mod(fibonacci(i^2)+fibonacci(2*i),i^2+2*i)==0,print1(i,",")))

Offset changed from 0 to 1 by Robert G. Wilson v, Nov 07 2010

More terms from Robert G. Wilson v, Nov 07 2010

A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

Original entry on oeis.org

5, 17, 617, 857, 1277, 1427, 1697, 2087, 2687, 3557, 4217, 5417, 5477, 7307, 8837, 9437, 10067, 13877, 17657, 18287, 20747, 21587, 23537, 25577, 27917, 28547, 30467, 32117, 32297, 35507, 37337, 37547, 40127, 41177, 41387, 41957

Offset: 1

Karl V. Keller, Jr., Jan 09 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n, n+2, n+24, n+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n) = 30*n + 17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
A253624 is a subsequence of this sequence.

			For n = 17, the numbers 17, 19, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A253624.

a245568[n_] := Select[Prime@ Range@ n, And[PrimeQ[# + 2], PrimeQ[# + 24], PrimeQ[# + 26]] &]; a245568[5000] (* Michael De Vlieger, Jan 11 2015 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+24) and isprime(n+26): print(n,end=', ')

A247089 Initial members of prime quadruples (p, p+2, p+30, p+32).

Original entry on oeis.org

11, 29, 41, 71, 107, 149, 197, 239, 281, 431, 569, 827, 1019, 1031, 1061, 1289, 1451, 1667, 1997, 2081, 2111, 2237, 2309, 2657, 2969, 3299, 3329, 3359, 3527, 3821, 4019, 4127, 4229, 4241, 4517, 5849, 6269, 6659, 6761, 7457, 7559, 8597

Offset: 1

Karl V. Keller, Jr., Jan 10 2015

nonn

Primes p such that (p, p+2) and (p+30, p+32) are twin prime pairs.
This sequence is a subsequence of A001359 (lesser of twin primes).
The subset of terms ending in 1 in this sequence is a subsequence of A132232 (primes, 11 mod 30).
The subset of terms ending in 7 in this sequence is a subsequence of A141860 (primes, 2 mod 15).
The subset of terms ending in 9 in this sequence is a subsequence of A132236 (primes, 29 mod 30).

			For n=11, the numbers 11, 13, 41, 43, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A001359, A132232, A132236, A141860, A181603 (twins, end 1), A181605 (twins, end 7), A181606 (twins, end 9).

a247089[n_] := Select[Prime@ Range@ n, And[PrimeQ[# + 2], PrimeQ[# + 30], PrimeQ[# + 32]] &]; a247089[1100] (* Michael De Vlieger, Jan 11 2015 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+30) and isprime(n+32): print(n,end=', ')

A248523 Initial members of prime quadruples (n, n+2, n+144, n+146).

Original entry on oeis.org

5, 137, 1787, 1997, 2237, 2657, 3527, 4127, 4337, 4787, 8087, 12107, 13757, 14447, 17987, 19697, 21377, 23057, 23687, 31247, 32297, 34157, 34367, 35447, 37547, 38567, 39227, 43397, 48677, 51197, 51827, 53087, 58907, 65027, 65837

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+144,n+146).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).

			For n=137, the numbers 137, 139, 281, 283, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

from sympy import isprime
for n in range(1,10000001,2):
    if isprime(n) and isprime(n+2) and isprime(n+144) and isprime(n+146): print(n,end=', ')

A248661 Initial members of prime quadruples (n, n+2, n+54, n+56).

Original entry on oeis.org

5, 17, 137, 227, 827, 1427, 1667, 1877, 2027, 2087, 2657, 3527, 3767, 4217, 4967, 10037, 11117, 11777, 12107, 13877, 17987, 19697, 20717, 21557, 22037, 23687, 24977, 27527, 27737, 34157, 37307, 41177, 42017, 42407, 47657, 48677

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+54,n+56).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (primes, 30n-13), A181605 (twin primes, end 7), and A092340 (prime n, where n^2+2*n divides (fibonacci(n^2)+fibonacci(2*n))).

			For n=17, the numbers 17, 19, 71, 73, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A092340.

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+54) and isprime(n+56): print(n,end=', ')

cp's OEIS Frontend

A181603 Twin primes ending in 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

A181606 Twin primes ending in 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A181604 Twin primes ending in 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A253624 Initial members of prime sextuples (p, p+2, p+12, p+14, p+24, p+26).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python

A092340 Prime numbers n such that n^2+2*n divides (Fibonacci(n^2) + Fibonacci(2*n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A092340 Prime numbers n such that n^2+2n divides (Fibonacci(n^2) + Fibonacci(2n)).