A023200 -id:A023200 - cp's OEIS frontend

A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

7, 13, 37, 97, 103, 223, 307, 457, 853, 877, 1087, 1297, 1423, 1483, 1867, 1993, 2683, 3457, 4513, 4783, 5227, 5647, 6823, 7873, 8287, 10453, 13687, 13873, 15727, 16057, 16063, 16183, 17383, 19417, 19423, 20743, 21013, 21313, 22273, 23053, 23557

Offset: 1

Labos Elemer, Mar 22 2000

nonn

The sequence includes A052166, A052168, A022008 and also other primes like 13, 103, 16063 etc.
a(n) is the lesser term of a 4-twin (A023200) after which the next 4-twin comes in minimal distance [here it is 2; see A052380(4/2)].
Analogous prime sequences are A047948, A052376, A052377 and A052188-A052198 with various [d, A052380(d/2), d] difference patterns following a(n).
All terms == 1 (mod 6) - Zak Seidov, Aug 27 2012
Subsequence of A022005. - R. J. Mathar, May 06 2017

			103 initiates [103,107,109,113] prime quadruple followed by [4,2,4] difference pattern.

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A023200, A053320, A022008, A052166, A052168, A001223, A052380.

a = {}; Do[If[Prime[x + 3] - Prime[x] == 10, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Zerinvary Lajos, Apr 03 2007 *)
Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&][[All,1]] (* Harvey P. Dale, Jun 16 2017 *)

is(n)=n%6==1 && isprime(n+4) && isprime(n+6) && isprime(n+10) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

a(n) is the initial prime of a [p, p+4, p+6, p+6+4] prime-quadruple consisting of two 4-twins: [p, p+4] and [p+6, p+10].

A087679 Numbers k such that both k+2 and k-2 are prime.

Original entry on oeis.org

5, 9, 15, 21, 39, 45, 69, 81, 99, 105, 111, 129, 165, 195, 225, 231, 279, 309, 315, 351, 381, 399, 441, 459, 465, 489, 501, 615, 645, 675, 741, 759, 771, 825, 855, 861, 879, 885, 909, 939, 969, 1011, 1089, 1095, 1215, 1281, 1299, 1305, 1425, 1431, 1449, 1485

Offset: 1

Zak Seidov, Sep 27 2003

Essentially the same as A029708: a(n) = A029708(n-1) for n>=2.
Midpoint of cousin prime pairs.
The only prime is 5. All other terms are multiples of 3. - Zak Seidov, May 19 2014

M. F. Hasler, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A014574, A087678-A087683, A087695-A087697, A088762.

ZL:=[]:for p from 1 to 1485 do if (isprime(p) and isprime(p+4) ) then ZL:=[op(ZL),(p+(p+4))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

lst={};Do[If[PrimeQ[n-2]&&PrimeQ[n+2],AppendTo[lst,n]],{n,3,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 14 2009 *)

s=[]; for(n=1, 2000, if(isprime(n-2) && isprime(n+2), s=concat(s, n))); s \\ Colin Barker, May 19 2014

is_A087679(n)={isprime(n-2) && isprime(n+2)} \\ For numbers >> 10^12 one should add conditions {n%6==3 && ... || n==5} or consider only such numbers congruent to 3 (mod 6). - M. F. Hasler, Apr 05 2017

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

a(n+1) = A056956(n)*6 + 3 = A157834(n)*3; a(n) = A088762(n)*2 + 1. - M. F. Hasler, Apr 05 2017

More terms from Ray Chandler, Oct 26 2003

A015913 Numbers k such that sigma(k) + 4 = sigma(k+4).

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423

Offset: 1

Robert G. Wilson v

nonn

This sequence contains the composite number 305635357, so is different from A023200 and A029710 (305635357 is the only composite member of the present sequence below 10^9). - Jud McCranie, Jan 07 2001

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

Cf. A023200, A029710.

Select[Range[1500],DivisorSigma[1,#]+4==DivisorSigma[1,#+4]&] (* Harvey P. Dale, Nov 04 2011 *)

is(n)=sigma(n)+4==sigma(n+4) \\ Charles R Greathouse IV, Mar 09 2014

A049488 Primes p such that p+16 is prime.

Original entry on oeis.org

3, 7, 13, 31, 37, 43, 67, 73, 97, 151, 157, 163, 181, 211, 223, 241, 277, 331, 337, 367, 373, 433, 463, 487, 541, 547, 571, 577, 601, 631, 643, 661, 727, 757, 811, 823, 937, 967, 997, 1033, 1087, 1093, 1171, 1201, 1213, 1291, 1303, 1423, 1471, 1483, 1543

Offset: 1

Labos Elemer

Using the Elliott-Halberstam conjecture, Goldston et al. prove that there are an infinite number of primes here. - T. D. Noe, Nov 26 2013

			7 and 7+16=23 are prime.

P. D. T. A. Elliott and H. Halberstam, A conjecture in prime number theory, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 59-72, Academic Press, London, 1970.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms up to a(1000) by T. D. Noe)
D. A. Goldston, J. Pintz, and C. Y. Yildirim, Primes in Tuples I, arXiv:math/0508185 [math.NT], Aug 10 2005.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], Aug 24 2004.

Cf. A001359, A023200, A023202.

Select[Range[1000], PrimeQ[#] && PrimeQ[#+16]&] (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
Select[Prime[Range[250]],PrimeQ[#+16 ]&] (* Harvey P. Dale, Oct 30 2015 *)

select(p->isprime(p+16),primes(100)) \\ Charles R Greathouse IV, Jul 08 2013

A172367 Numbers k > 0 such that k+4 is a prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 25, 27, 33, 37, 39, 43, 49, 55, 57, 63, 67, 69, 75, 79, 85, 93, 97, 99, 103, 105, 109, 123, 127, 133, 135, 145, 147, 153, 159, 163, 169, 175, 177, 187, 189, 193, 195, 207, 219, 223, 225, 229, 235, 237, 247, 253, 259, 265, 267, 273, 277, 279

Offset: 1

Juri-Stepan Gerasimov, Feb 01 2010

The subsequence of primes A023200 consists of the smallest primes p of cousin prime pairs (p, p+4), while the subsequence of nonprimes is A164384. - Bernard Schott, Oct 19 2021

			a(1) = 5 - 4 = 1, a(2) = 7 - 4 = 3.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Union of A023200 and A164384.

Cf. A000040, A006093, A040976, A086801.

Table[Prime[n]-4,{n,3,53}] (* Charles R Greathouse IV, Mar 12 2012 *)

a(n)=prime(n+2)-4 \\ Charles R Greathouse IV, Mar 12 2012

a(n) = prime(n+2) - 4.

A049492 Primes p such that p+4 and p+16 are also primes.

Original entry on oeis.org

3, 7, 13, 37, 43, 67, 97, 163, 223, 277, 463, 487, 643, 757, 823, 937, 967, 1087, 1093, 1213, 1303, 1423, 1483, 1567, 1597, 1693, 1873, 2083, 2137, 2293, 2377, 2617, 2683, 2953, 3187, 3343, 3847, 3907, 4003, 4447, 4783, 5503, 5653, 5923, 6547, 6967, 6997

Offset: 1

Labos Elemer

nonn

All terms > 3 are == 1 (mod 6). - Zak Seidov, Sep 05 2014

Intersection of A023200 and A049488. - Michel Marcus, Sep 05 2014

			3, 3+4 = 7, 3+16 = 19 are all primes.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A023200, A049488, A049493.

Select[Prime[Range[900]],And@@PrimeQ[#+{4,16}]&] (* Harvey P. Dale, Jan 17 2011 *)

lista(nn) = forprime (n=1, nn, if (isprime(n+4) && isprime(n+16), print1(n, ", "))); \\ Michel Marcus, Sep 05 2014

A074822 Primes p such that p + 4 is prime and p == 9 (mod 10).

Original entry on oeis.org

19, 79, 109, 229, 349, 379, 439, 499, 739, 769, 859, 1009, 1279, 1429, 1489, 1549, 1579, 1609, 1999, 2239, 2269, 2389, 2539, 2659, 2689, 2749, 3019, 3079, 3319, 3529, 3919, 4129, 4519, 4639, 4729, 4789, 4969, 4999, 5479, 5569, 5689, 5779, 5839, 6199

Offset: 1

Roger L. Bagula, Sep 30 2002

From Rémi Eismann, May 14 2006; May 04 2007: (Start)
Also primes for which k is equal to 5 in A117078. Examples: prime(9) = prime(8) + (prime(8) mod 5) = 19 + (19 mod 5)=23; prime(23) = prime(22) + (prime(22) mod 5) = 79 + (79 mod 5)=83; prime(1359) = prime(1358) + (prime(1358) mod 5) = 11239+ (11239 mod 5)=11243.
The prime numbers in this sequence are of the form (10i-1) with i=(level(n)+1)/2, level(n) defined in A117563.
Consider A117078: a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists. Sequence gives values of prime(n) for which k=5. (End)
p is the lesser member of cousin primes (p,p+4) such that p == 9 (mod 10). - Muniru A Asiru, Jul 03 2017

Remi Eismann, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A001223, A117078, A117563.

Intersection of A023200 and A030433.

Prime[ Select[ Range[1000], Prime[ # ] + 4 == Prime[ # + 1] && Mod[ Prime[ # ], 10] == 9 & ]]
Transpose[Select[Partition[Prime[Range[820]],2,1],Last[#]-First[#] == 4 && Mod[ First[ #],10]==9&]][[1]] (* Harvey P. Dale, Oct 20 2011 *)

is(n)=n%30==19 && isprime(n+4) && isprime(n) \\ Charles R Greathouse IV, Jul 12 2017

list(lim)=my(v=List(),p=19); forprime(q=23,lim+4, if(q-p==4 && p%30==19, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jul 12 2017

Edited by Robert G. Wilson v and N. J. A. Sloane, Oct 03 2002

Entry revised by N. J. A. Sloane, Feb 24 2007

A056956 Numbers n such that 6n+1 and 6n+5 are both primes.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 13, 16, 17, 18, 21, 27, 32, 37, 38, 46, 51, 52, 58, 63, 66, 73, 76, 77, 81, 83, 102, 107, 112, 123, 126, 128, 137, 142, 143, 146, 147, 151, 156, 161, 168, 181, 182, 202, 213, 216, 217, 237, 238, 241, 247, 248, 258, 261, 263, 266, 268, 277, 282

Offset: 1

Henry Bottomley, Jul 18 2000

nonn

Note that if prime p>3 then p mod 6 = 1 or 5.

			a(2)=2 since 6*2+1=13 and 6*2+5=17 are both prime.

M. F. Hasler, Table of n, a(n) for n = 1..5000

Cf. A002822, A024898, A024899, A059325.

Select[Range[300], And @@ PrimeQ /@ ({1, 5} + 6#) &] (* Ray Chandler, Jun 29 2008 *)

is(n)=isprime(n*6+1)&&isprime(n*6+5) \\ M. F. Hasler, Apr 05 2017

a(n) = (A023200(n+1)-1)/6 = (A046132(n+1)-5)/6 = A047847(n+1)/3

a(n) = floor(A087679(n+1)/6). - M. F. Hasler, Apr 05 2017

Edited by N. J. A. Sloane, Nov 07 2006

A046136 Primes p such that p, p+4 and p+10 are primes.

Original entry on oeis.org

3, 7, 13, 19, 37, 43, 79, 97, 103, 127, 163, 223, 229, 307, 349, 379, 439, 457, 499, 643, 673, 853, 877, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1423, 1429, 1483, 1489, 1549, 1597, 1609, 1867, 1993, 2203, 2347, 2389, 2437, 2539, 2683, 2689, 2833, 2953

Offset: 1

Eric W. Weisstein

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A023200, A023203.

q:= p-> andmap(isprime, [p, p+4, p+10]):
select(q, [$2..3000])[];  # Alois P. Heinz, Feb 23 2020

Select[Range@ 2820, AllTrue[{#, # + 4, # + 10}, PrimeQ] &] (* Michael De Vlieger, Jul 24 2015, Version 10 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+4) && isprime(p+10), print1(p, ", "))); \\ Michel Marcus, Jul 24 2015

A023200 INTERSECT A023203. - R. J. Mathar, Jan 23 2009

A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

Original entry on oeis.org

434, 305635357, 104, 27, 195556, 65, 12, 39, 20, 56, 916, 80, 212282, 57, 44, 106645, 52, 125

Offset: 1

Labos Elemer May 23 2000

a(19) > 4293000000, if it exists. - Jud McCranie, May 25 2000

a(19) > 10^11, if it exists. - Charles R Greathouse IV, Oct 26 2022

			a(5) corresponds to n=3+2=5, d=2n=10 and the smallest composite integer is 195556. The next solution is 1152136225.

Mauro Fiorentini, Le soluzioni dell’equazione s(n + k) = s(n) + k, con n composto fino a 109 e k sino a 1000.

Cf. A015913, A015914, A015915, A015916, A015917, A054799.

Cf. A023200, A023201, A023202, A023203,

a(n)=forcomposite(x=3,10^66,if(sigma(x)+2*n==sigma(x+2*n),return(x)));
for(n=1,66,print1(a(n),", ")); \\ Joerg Arndt, Nov 15 2014

a19(lim,startAt=39)=startAt=ceil(startAt); my(v=vectorsmall(38),i=(startAt-1)%38); forfactored(n=startAt,lim\1+38, my(t=sigma(n)); if(i++>38,i=1); if(t==v[i]+38, return(n[1]-38)); v[i]=if(vecsum(n[2][,2])>1,t,0)) \\ Charles R Greathouse IV, Oct 25 2022

Description corrected by Jud McCranie, May 25 2000

cp's OEIS Frontend

A052378 Primes followed by a [4,2,4] prime difference pattern of A001223.

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A087679 Numbers k such that both k+2 and k-2 are prime.

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A015913 Numbers k such that sigma(k) + 4 = sigma(k+4).

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A049488 Primes p such that p+16 is prime.

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A172367 Numbers k > 0 such that k+4 is a prime.

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A049492 Primes p such that p+4 and p+16 are also primes.

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A074822 Primes p such that p + 4 is prime and p == 9 (mod 10).

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