A023200 -id:A023200 - cp's OEIS frontend

A103195 Concatenations of pairs of primes that differ by four.

37, 711, 1317, 1923, 3741, 4347, 6771, 7983, 97101, 103107, 109113, 127131, 163167, 193197, 223227, 229233, 277281, 307311, 313317, 349353, 379383, 397401, 439443, 457461, 463467, 487491, 499503, 613617, 643647, 673677, 739743, 757761, 769773

Offset: 1

Parthasarathy Nambi, Mar 18 2005

			The primes 3 and 7 differ by four, so the first term is 37.

Chris Caldwell, The First 1,000 Primes.
Eric Weisstein's World of Mathematics, Cousin Primes.
M. Wolf, On Twin and Cousin Primes.

s = Select[ Prime[ Range[ 140]], PrimeQ[ # + 4] &]; FromDigits /@ Join @@@ IntegerDigits /@ Transpose[{s, s + 4}] (* Robert G. Wilson v, Mar 19 2005 *)
Join[{37},FromDigits[Flatten[IntegerDigits/@#]]&/@Select[Partition[ Prime[ Range[ 200]],2,1],#[[2]]-#[[1]]==4&]] (* Harvey P. Dale, Sep 26 2016 *)

a(n) = A023200(n) concatenated with A023200(n)+4. - Jonathan Vos Post, Mar 19 2005

More terms from Robert G. Wilson v, Mar 19 2005

A173037 Numbers k such that k-4, k-2, k+2 and k+4 are prime.

Original entry on oeis.org

9, 15, 105, 195, 825, 1485, 1875, 2085, 3255, 3465, 5655, 9435, 13005, 15645, 15735, 16065, 18045, 18915, 19425, 21015, 22275, 25305, 31725, 34845, 43785, 51345, 55335, 62985, 67215, 69495, 72225, 77265, 79695, 81045, 82725, 88815, 97845

Offset: 1

Juri-Stepan Gerasimov, Feb 07 2010

nonn

Average k of the four primes in two twin prime pairs (k-4, k-2) and (k+2, k+4) which are linked by the cousin prime pair (k-2, k+2).
All terms are odd composites; except for a(1) they are multiples of 5.
Subsequence of A087679, of A087680 and of A164385.
All terms except for a(1) are multiples of 15. - Zak Seidov, May 18 2014
One of (k-1, k, k+1) is always divisible by 7. - Fred Daniel Kline, Sep 24 2015
Terms other than a(1) must be equivalent to 1 mod 2, 0 mod 3, 0 mod 5, and 0,+/-1 mod 7. Taken together, this requires terms other than a(1) to have the form 210k+/-15 or 210k+105. However, not all numbers of that form belong to this sequence. - Keith Backman, Nov 09 2023

			9 is a term because 9-4 = 5 is prime, 9-2 = 7 is prime, 9+2 = 11 is prime and 9+4 = 13 is prime.

Klaus Brockhaus, Table of n, a(n) for n = 1..28388 (terms < 10^9).

Cf. A001359, A006512, A007530, A007811, A014561, A023200, A038800, A046132, A087680, A112540, A125855, A164385.

[ p+4: p in PrimesUpTo(100000) | IsPrime(p) and IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8) ]; // Klaus Brockhaus, Feb 09 2010

Select[Range[100000],AllTrue[#+{4,2,-2,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 30 2015 *)

is(n)=isprime(n-4) && isprime(n-2) && isprime(n+2) && isprime(n+4) \\ Charles R Greathouse IV, Sep 24 2015

from sympy import primerange
def aupto(limit):
    p, q, r, alst = 2, 3, 5, []
    for s in primerange(7, limit+5):
        if p+2 == q and p+6 == r and p+8 == s: alst.append(p+4)
        p, q, r = q, r, s
    return alst
print(aupto(10**5)) # Michael S. Branicky, Feb 03 2022

For n >= 2, a(n) = 15*A112540(n-1). - Michel Marcus, May 19 2014
From Jeppe Stig Nielsen, Feb 18 2020: (Start)
For n >= 2, a(n) = 30*A014561(n-1) + 15.
For n >= 2, a(n) = 10*A007811(n-1) + 5.
a(n) = A007530(n) + 4.
a(n) = A125855(n) + 5. (End)

Edited and extended beyond a(9) by Klaus Brockhaus, Feb 09 2010

A186243 Numbers k such that 6k-5 and 6k-1 are both primes.

Original entry on oeis.org

2, 3, 4, 7, 8, 12, 14, 17, 18, 19, 22, 28, 33, 38, 39, 47, 52, 53, 59, 64, 67, 74, 77, 78, 82, 84, 103, 108, 113, 124, 127, 129, 138, 143, 144, 147, 148, 152, 157, 162, 169, 182, 183, 203, 214, 217, 218, 238, 239, 242, 248, 249, 259, 262, 264, 267, 269

Offset: 1

Jonathan Vos Post, Feb 15 2011

Numbers k such that 6*k-5 and 6*k-1 are cousin primes. The D = 2 numbers in class II, from page 3 of Weber. - Jonathan Vos Post, Feb 14 2011

			a(3) = 4 because 6*4-5 = 19 is prime, and 6*4-1 = 23 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000
H. J. Weber, Exceptional Prime Number Twins, Triplets and Multiplets, arXiv:1102.3075 [math.NT], 2011.
Eric W. Weisstein, Cousin Primes.

Cf. A023200, A046132, A088765.

Select[Range[400], PrimeQ[6#-5] && PrimeQ[6#-1] &] (* Alonso del Arte, Feb 16 2011 *)

{k such that 6*k-5 is in A023200} = {k such that 6*k-1 is in A046132}.

A237890 Primes p such that p^2 + 4 and p^2 + 10 are also primes.

Original entry on oeis.org

3, 7, 13, 97, 487, 613, 743, 827, 883, 1117, 1987, 2477, 2887, 3863, 4483, 5153, 5557, 5683, 5923, 5953, 6287, 7643, 7937, 8093, 9323, 10343, 12377, 13033, 13063, 14087, 14767, 15373, 16937, 17713, 17987, 18257, 19013, 19333, 19753, 19853, 20287, 20873, 21673

Offset: 1

K. D. Bajpai, Feb 15 2014

nonn

			7 is prime and appears in the sequence because 7^2+4 = 53 and 7^2+10 = 59 are also primes.
97 is prime and appears in the sequence because 97^2+4 = 9413 and 97^2+10 = 9419 are also primes.

K. D. Bajpai, Table of n, a(n) for n = 1..1300

Cf. A000040, A023200, A046136, A230223.

KD := proc() local a,b,d;  a:=ithprime(n);  b:=a^2+4; d:=a^2+10;  if isprime (b) and isprime(d) then RETURN (a); fi;  end: seq(KD(), n=1..5000);

Select[Prime[Range[5000]], PrimeQ[#^2 + 4] && PrimeQ[#^2 + 10] &]

s=[]; forprime(p=2, 25000, if(isprime(p^2+4) && isprime(p^2+10), s=concat(s, p))); s \\ Colin Barker, Feb 15 2014

A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

Original entry on oeis.org

3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13

Offset: 1

Labos Elemer, May 23 2000

nonn

Least (prime) solutions for phi(x+2n)=phi(x)+2n seems to be identical to this sequence, while prime solutions are indeed identical to this sequence.
2nd definition = smallest number x such that phi(x+2n)=phi(x)+2n.
3rd definition = smallest primes p such that p+2n=q prime (A020483).
The 3 definitions are identical or conjectured to be identical.
The definitions are not identical if we do not take the smallest numbers. These smallest solutions are believed to be always prime numbers.
Duplicate of A020483, assuming that the 3rd definition is also correct. - R. J. Mathar, Apr 26 2015
If it can be proved that all these definitions are identical, then this entry should be merged with A020483. - N. J. A. Sloane, Feb 06 2017

			n-th primes 2,3,5,7,11,13, are solutions to sigma(x+2n)=2n+sigma(x) at 2n=2,6,22,116,88.

Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions. Marcel Dekker,Inc., New York.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A023200-A023203, A015913-A015917, A000203, A000010, A020483.

A054906 := proc(n)
    local x;
    for x from 0 do
        if numtheory[sigma](x+2*n) = numtheory[sigma](x)+2*n then
            return x;
        end if;
    end do:
end proc:
seq(A054906(n),n=1..40); # R. J. Mathar, Sep 23 2016

Table[x = 1; While[DivisorSigma[1, x + 2 n] != DivisorSigma[1, x] + 2 n, x++]; x, {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

a(n) = my(x = 1); while(sigma(x+2*n) != sigma(x)+2*n, x++); x; \\ Michel Marcus, Dec 17 2013

Minimal solutions to A000203(x+2n)=A000203(x)+2n or to A000010(x+2n)=A000010(x)+2n or to p+2n=q; p, q primes, a(n)=p.

a(n) <= A054905(n). - R. J. Mathar, Apr 28 2015

A088762 Numbers n such that (2n-1, 2n+3) is a cousin prime pair.

Original entry on oeis.org

2, 4, 7, 10, 19, 22, 34, 40, 49, 52, 55, 64, 82, 97, 112, 115, 139, 154, 157, 175, 190, 199, 220, 229, 232, 244, 250, 307, 322, 337, 370, 379, 385, 412, 427, 430, 439, 442, 454, 469, 484, 505, 544, 547, 607, 640, 649, 652, 712, 715, 724, 742, 745, 775, 784, 790

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..4200
Eric Weisstein's World of Mathematics, Cousin Primes

Essentially the same as A111981.

Cf. A087679, A088764, A088766, A088768, A088770.

[n: n in [1..1000] |IsPrime(2*n-1) and IsPrime(2*n+3)]; // Vincenzo Librandi, May 19 2017

Select[Range[2000],PrimeQ[2*#-1]&&PrimeQ[2*#+3]&] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

a(n) = (A087679(n)-1)/2 = (A023200(n)+1)/2 = (A046132(n)-3)/2.

A164572 Numbers k such that k and k+4 are both prime powers.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 13, 19, 23, 25, 27, 37, 43, 49, 67, 79, 97, 103, 109, 121, 127, 163, 169, 193, 223, 229, 239, 277, 289, 307, 313, 343, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 729, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087

Offset: 1

Daniel Forgues, Aug 16 2009, Aug 17 2009

nonn

Numbers n such that n + (0, 4) is a prime power pair.
A generalization of the cousin primes. The cousin primes are a subsequence.
n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.
n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1256 from Daniel Forgues)

Cf. A023200, A000961.

k and (x) are prime powers: A006549 (k+1), A120431 (k+2), A164571 (k+3), this sequence (k+4), A164573 (k+5), A164574 (k+6).

Select[Range[1000], PrimeNu[#] < 2 && PrimeNu[# + 4] < 2 &] (* Amiram Eldar, Oct 01 2020 *)

is(n)=if(n==1,return(1)); isprimepower(n) && isprimepower(n+4) \\ Charles R Greathouse IV, Apr 24 2015

A221211 Numbers n such that n and n + 4 are prime and there is a power of two in the interval (n,n+4).

Original entry on oeis.org

3, 7, 13, 127

Offset: 1

Brad Clardy, Feb 21 2013

nonn

It is a conjecture that this is a finite sequence. These may be the only known cousin primes with this property.

The Cf.s list similar sequences, of the form -- numbers n such that n and n+m are prime and contain a power of two in the interval (n,n+m). The case where m=2, the twin prime case -- not listed, has only one member n=3. Another member would have to be a twin where n+2 was a Fermat type prime and n a Mersenne prime.

Cf. A023200.

Cf. A220951 (gap of 6), A213210 (8), A220746 (10), A213677 (12), A222424 (14), A222227 (16), A222219 (18).

//Program finds primes separated by an even number (called gap) which
//have a power of two between them. The program starts with the smallest
//of two above gap. Primes less than this starting point can be checked by
//inspection. In this example 3 also works.
gap:=4;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
    powerof2:=2^i;
    for k:=powerof2-gap+1 to powerof2-1 by 2 do
        if (IsPrime(k) and IsPrime(k+gap)) then k;
        end if;
    end for;
end for;

[n: n in PrimesUpTo(10^3) | IsPrime(n+4) and exists{t: t in [n+1..n+3 by 2] | IsOne(t/2^Valuation(t,2))}]; // Bruno Berselli, May 16 2013

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 2, 12, 39, 98, 526, 2117, 6663, 15043, 68403, 791581, 4826577, 19592777, 102551299, 739788968, 4449585790, 36547266589, 324446266072, 2743681178070

Offset: 1

Zhi-Wei Sun, Sep 08 2013

Conjecture: a(n) > 0 for all n > 0.
This implies the twin prime conjecture, and it is similar to the prime circle problem mentioned in A051252.
For each n = 2,3,... construct an undirected simple graph T(n) with vertices 0,1,...,n which has an edge connecting two distinct vertices i and j if and only if 6*(i+j)-1 and 6*(i+j)+1 are twin primes. Then a(n) is just the number of Hamiltonian cycles contained in T(n). Thus a(n) > 0 if and only if T(n) is a Hamilton graph.
Zhi-Wei Sun also made the following similar conjectures for odd primes, Sophie Germain primes, cousin primes and sexy primes:
(1) For any integer n > 0, there is a permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p-1)/2, where p is an odd prime. Also, we may replace the above (p-1)/2 by (p+1)/4 or (p-1)/6; when n > 4 we may substitute (p-1)/4 for (p-1)/2.
(2) For any integer n > 2, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p+1)/6, where p is a Sophie Germain prime.
(3) For any integer n > 3, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 6*k+1 and 6*k+5 both prime.
(4) For any integer n > 4, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 2*k-3 and 2*k+3 both prime.

			a(n) = 1 for n = 1,2,3 due to the permutation (0,...,n).
a(4) = 2 due to the permutations (0,1,4,3,2) and (0,2,1,4,3).
a(5) = 2 due to the permutations (0,1,4,3,2,5), (0,3,4,1,2,5).
a(6) = 2 due to the permutations
  (0,1,6,4,3,2,5) and (0,3,4,6,1,2,5).
a(7) = 5 due to the permutations
  (0,1,6,4,3,2,5,7), (0,1,6,4,3,7,5,2), (0,2,1,6,4,3,7,5),
  (0,3,4,6,1,2,5,7), (0,5,2,1,6,4,3,7).
a(8) = 2 due to the permutations
  (0,1,6,4,8,2,3,7,5) and (0,1,6,4,8,2,5,7,3).
a(9) = 12 due to the permutations
  (0,1,6,4,3,9,8,2,5,7), (0,1,6,4,8,9,3,2,5,7),
  (0,1,6,4,8,9,3,7,5,2), (0,2,1,6,4,8,9,3,7,5),
  (0,2,8,9,1,6,4,3,7,5), (0,3,4,6,1,9,8,2,5,7),
  (0,3,9,1,6,4,8,2,5,7), (0,3,9,8,4,6,1,2,5,7),
  (0,5,2,1,6,4,8,9,3,7), (0,5,2,8,4,6,1,9,3,7),
  (0,5,2,8,9,1,6,4,3,7), (0,5,7,3,9,1,6,4,8,2).
a(10) > 0 due to the permutation (0,5,2,3,9,1,6,4,8,10,7).
a(11) > 0 due to the permutation (0,10,8,9,3,7,11,6,4,1,2,5).
a(12) > 0 due to the permutation
        (0, 5, 2, 1, 6, 4, 3, 9, 8, 10, 7, 11, 12).

Zhi-Wei Sun, Twin primes and circular permutations, a message to Number Theory List, Sept. 8, 2013.
Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.

Cf. A000040, A001359, A006512, A023200, A046132, A023201, A046117, A005384, A051252, A228766, A228860, A228886.

(* A program to compute required circular permutations for n = 7. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0,7,5,2,3,4,6,1) is identical to (0,1,6,4,3,2,5,7) if we ignore direction. Thus a(7) is half of the number of circular permutations yielded by this program. *)
tp[n_]:=tp[n]=PrimeQ[6n-1]&&PrimeQ[6n+1]
V[i_]:=Part[Permutations[{1,2,3,4,5,6,7}],i]
m=0
Do[Do[If[tp[If[j==0,0,Part[V[i],j]]+If[j<7,Part[V[i],j+1],0]]==False,Goto[aa]],{j,0,7}];
m=m+1;Print[m,":"," ",0," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]];Label[aa];Continue,{i,1,7!}]

a(10)-a(25) from Max Alekseyev, Sep 12 2013

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

Original entry on oeis.org

1, 1, 1, 1, 3556, 1, 34, 3, 4, 1, 2, 1, 11285, 5, 2, 124, 569, 1, 290, 3, 1, 165, 2, 1, 1, 2, 1, 316, 1, 2, 58957, 1, 3, 58617, 522, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 7932, 4, 1, 5875, 1679, 4, 4, 3, 3, 1, 2, 307, 1, 1, 1, 1, 1, 4, 3206, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 2, 11170, 1, 2, 4245, 1, 1, 81, 2, 1, 1, 2, 58, 1, 3, 4, 7303, 1, 1, 5, 1, 3, 3, 3, 383, 111408, 1

Offset: 0

Farideh Firoozbakht and Jahangeer Kholdi, Nov 25 2014

nonn

All terms conjecturally are found. Note that according to the definition a(k,0) is the k-th term of the sequence {p | p is prime} namely for every positive integer k, a(k,0) = prime(k). Hence if Firoozbakht's conjecture is true then a(0)=1.

			a(0)=a(1)=a(2)=a(3)=1 conjecturally states that the four sequences A000040, A001359, A023200 and A023201 have this property: For every positive integer n, b(n) exists and b(n+1) < b(n)^(1+1/n). Namely b(n)^(1/n) is a strictly decreasing function of n.
If in the definition instead of the sequence {p | p and p+2n are primes} we set {p | p is prime and nextprime(p)=p+2n} then it seems that except for n=3 all terms of the new sequence {c(n)} are equal to 1 and for n=3, c(3)=7746. Note that c(3)=7746 means that the sequence {p | p is prime and nextprime(p)=p+6} = A031924 has this property: For all k >= 7746, A031924(k+1)^(1/(k+1)) < A031924(k)^(1/k).

Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001359, A023200, A023201, A023202, A023203, A031924.

cp's OEIS Frontend

A103195 Concatenations of pairs of primes that differ by four.

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A173037 Numbers k such that k-4, k-2, k+2 and k+4 are prime.

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Magma

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A186243 Numbers k such that 6*k-5 and 6*k-1 are both primes.

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A237890 Primes p such that p^2 + 4 and p^2 + 10 are also primes.

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Maple

Mathematica

PARI

A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

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Maple

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A088762 Numbers n such that (2n-1, 2n+3) is a cousin prime pair.

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Magma

Mathematica

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A164572 Numbers k such that k and k+4 are both prime powers.

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A186243 Numbers k such that 6k-5 and 6k-1 are both primes.

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.