A060230 -id:A060230 - cp's OEIS frontend

A060229 Smaller member of a twin prime pair whose mean is a multiple of A002110(3)=30.

29, 59, 149, 179, 239, 269, 419, 569, 599, 659, 809, 1019, 1049, 1229, 1289, 1319, 1619, 1949, 2129, 2309, 2339, 2549, 2729, 2789, 2969, 2999, 3119, 3299, 3329, 3359, 3389, 3539, 3929, 4019, 4049, 4229, 4259, 4649, 4799, 5009, 5099, 5279, 5519, 5639

Offset: 1

Labos Elemer, Mar 21 2001

nonn

Equivalently, smaller of twin prime pair with primes in different decades.
Primes p such that p and p+2 are prime factors of Fibonacci(p-1) and Fibonacci(p+1) respectively. - Michel Lagneau, Jul 13 2016
The union of this sequence and A282326 gives A132243. - Martin Renner, Feb 11 2017
The union of {3,5}, A282321, A282323 and this sequence gives A001359. - Martin Renner, Feb 11 2017
The union of {3,5,7}, A282321, A282322, A282323, A282324, this sequence and A282326 gives A001097. - Martin Renner, Feb 11 2017
Number of terms less than 10^k, k=2,3,4,...: 2, 11, 72, 407, 2697, 19507, 146516, ... - Muniru A Asiru, Jan 29 2018

			For the pair {149,151} (149 + 151)/2 = 5*30.

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

Cf. A001359, A002110, A060230, A060231, A158277, A158861.

Subset of A001097, A001359, A132236, A132243 and A132247.

Filtered(List([0..200], k -> 30*k-1), 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 29 ]; // Vincenzo Librandi, Feb 13 2017

isA060229 := proc(n)
    if modp(n+1,30) =0 and isprime(n) and isprime(n+2) then
        true;
    else
        false;
    end if;
end proc:
A060229 := proc(n)
    option remember;
    if n =1 then
        29;
    else
        for a from procname(n-1)+2 by 2 do
            if isA060229(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A060229(n),n=1..80) ; # R. J. Mathar, Feb 19 2017

Select[Prime@ Range[10^3], PrimeQ[# + 2] && Mod[# + 1, 30] == 0 &] (* Michael De Vlieger, Jul 14 2016 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 30); \\ Michel Marcus, Dec 11 2013

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

A060232 Smaller of twin primes whose mean (average) is a multiple of A002110(6)=30030.

Original entry on oeis.org

180179, 270269, 300299, 330329, 390389, 420419, 540539, 660659, 840839, 1231229, 1261259, 1501499, 1621619, 1861859, 1921919, 1951949, 2012009, 2372369, 2762759, 2972969, 3663659, 3693689, 3723719, 3903899, 4084079

Offset: 1

Labos Elemer, Mar 21 2001

nonn

			For the pair {5735729,5735731} (5735729+5735731)/2 = 191*30030.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 642 terms from Muniru A Asiru)

Cf. A001359, A002110, A060229, A060230, A060231.

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 30030 = 0), k -> k[1]); # Muniru A Asiru, Jan 29 2018

for n from 1 to 10^5 do if (ithprime(n+1) - ithprime(n)) = 2 and (ithprime(n+1) + ithprime(n)) mod 30030 = 0 then print(ithprime(n)); fi; od; # Muniru A Asiru, Jan 29 2018
# More efficient:
select(t -> isprime(t) and isprime(t+2), [seq(30030*k-1, k=1..10^3)]); # Robert Israel, Jan 29 2018

Select[Partition[Prime[Range[300000]],2,1],#[[2]]-#[[1]]==2&&Divisible[Mean[ #],30030]&][[All,1]] (* Harvey P. Dale, Apr 23 2022 *)

Minor edits by Ray Chandler, Apr 04 2009

Definition clarified by Harvey P. Dale, Apr 23 2022

A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.

Original entry on oeis.org

3, 5, 29, 419, 2309, 180179, 4084079, 106696589, 892371479, 103515091679, 4412330782859, 29682952539239, 22514519501013539, 313986271960080719, 22750921955774182169, 912496437361321252439, 26918644902158976946979, 1290172194953476680815969, 1901713815361424627522739779

Offset: 1

Labos Elemer, Mar 22 2001

nonn

a(349) has 1001 digits. - Michael S. Branicky, Apr 19 2025

			a(13) = -1 + (2*3*5*7*...*41)*k(13) = 304250263527210*74 and {22514519501013539, 22514519501013542} are the corresponding primes; k(13)=74 is the smallest suitable multiplier. Twin primes obtained from primorial numbers with k=1 multiplier seem to be much rarer (see A057706).
For j=1,2,3,4,5,6, a(j)=A001359(1), A059960(1), A060229(1), A060230(1), A060231(1), A060232(1) respectively.

Michael S. Branicky, Table of n, a(n) for n = 1..348

Cf. A001359, A002110, A006794, A014545, A057706, A059960, A060229, A060230, A060231, A060232.

a(n) = {my(q = prod(k=1, n, prime(k))); for(k=1, oo, if (isprime(q*k-1) && isprime(q*k+1), return(q*k-1)););} \\ Michel Marcus, Jul 10 2018

from itertools import count
from sympy import primorial, isprime
def a(n):
    p = primorial(n)
    return next(m-1 for m in count(p, p) if isprime(m-1) and isprime(m+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 18 2025

a(n) = p = k(n)*q(n)-1, where q(n)=A002110(n) and k(n)=A060256(n) is the smallest integer whose multiplication by the n-th primorial yields p+1.

a(2)=5 corrected by Ray Chandler, Apr 03 2009

a(18) and beyond from Michael S. Branicky, Apr 18 2025

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A060229 Smaller member of a twin prime pair whose mean is a multiple of A002110(3)=30.

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

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A060232 Smaller of twin primes whose mean (average) is a multiple of A002110(6)=30030.

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GAP

Maple

Mathematica

Extensions

A060255 Smaller of twin primes {p, p+2} whose average p+1 = k*q is the least multiple of the n-th primorial number q such that k*q-1 and k*q+1 are twin primes.

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A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.