A124588 -id:A124588 - cp's OEIS frontend

A029710 Primes such that next prime is 4 greater.

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, 1429

Offset: 1

N. J. A. Sloane

nonn

Union with A124588 gives A124589. - Reinhard Zumkeller, Dec 23 2006
For any prime p > 3, if p + 4 is prime then necessarily it is the next prime. But there cannot be three consecutive primes with mutual distance 4: If p and p + 4 are prime, then p+8 is an odd multiple of 3 (cf. formula). - M. F. Hasler, Jan 15 2013
The smaller members p of cousin prime pairs (p,p+4) excluding p=3. - Marc Morgenegg, Apr 19 2016

			79 is a term as the next prime is 79 + 4 = 83. 3 is not a term even though 3 + 4 = 7 is prime, since it is not the next one.

Marius A. Burtea, Table of n, a(n) for n = 1..14741 ( first 1000 terms from R. Zumkeller )
Index entries for primes, gaps between

Essentially the same as A023200.

Cf. A001359, A029708, A031505, A124588, A124589, A157834.

p=primes(1700);m=1;
for u=1:length(p)-4
   if and(isprime(p(u)+4)==1,p(u+1)==p(u)+4);sol(m)=p(u);m=m+1;end
end
sol % Marius A. Burtea, Jan 24 2019

[p:p in PrimesUpTo(1700)| IsPrime(p+4) and NextPrime(p) eq p+4] // Marius A. Burtea, Jan 24 2019

for i from 1 to 226 do if ithprime(i+1) = ithprime(i) + 4 then print({ithprime(i)}); fi; od; # Zerinvary Lajos, Mar 19 2007

Select[Prime[Range[225]], NextPrime[#] == # + 4 &] (* Alonso del Arte, Jan 17 2013 *)
Transpose[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==4&]] [[1]] (* Harvey P. Dale, Mar 28 2016 *)

forprime(p=1, 1e4, if(nextprime(p+1)-p==4, print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

a(n) = A031505(n + 1) - 4 = A029708(n) - 2.

a(n) = 1 (mod 6) for all n; (a(n) + 2)/3 = A157834(n), i.e., a(n) = 3*A157834(n) - 2. - M. F. Hasler, Jan 15 2013

A124589 Primes p such that q-p <= 4, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 59, 67, 71, 79, 97, 101, 103, 107, 109, 127, 137, 149, 163, 179, 191, 193, 197, 223, 227, 229, 239, 269, 277, 281, 307, 311, 313, 347, 349, 379, 397, 419, 431, 439, 457, 461, 463, 487, 499, 521, 569, 599, 613, 617, 641, 643, 659

Offset: 1

N. J. A. Sloane, Dec 19 2006

nonn

Union of A124588 and A029710; complement of A124582. - Reinhard Zumkeller, Dec 23 2006

R. Zumkeller, Table of n, a(n) for n = 1..1000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

CF. A029710, A124582, A124588.

Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]<5&]][[1]] (* Harvey P. Dale, Apr 22 2013 *)

is(n)=isprime(n) && (isprime(n+2) || isprime(n+4) || n==2) \\ Charles R Greathouse IV, Jun 01 2016

a(n) >> n log^2 n. Infinite under standard conjectures. - Charles R Greathouse IV, Jun 01 2016

A192175 Array of primes determined by distance to next prime, by antidiagonals.

Original entry on oeis.org

2, 3, 7, 5, 13, 23, 11, 19, 31, 89, 17, 37, 47, 359, 139, 29, 43, 53, 389, 181, 199, 41, 67, 61, 401, 241, 211, 113, 59, 79, 73, 449, 283, 467, 293, 1831, 71, 97, 83, 479, 337, 509, 317, 1933, 523, 101, 103, 131, 491, 409, 619, 773, 2113, 1069, 887, 107

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1:  primes p such that p+1 or p+2 is a prime.
Row r>1:  primes p such that the least h for which p+2h is prime is r.
Rows 1-7:  A124588, A023200, A031924, A031926, A031928, A031932, A031924.

			Northwest corner:
  2.....3.....5.....11....17....29....41
  7.....13....19....37....43....67....79
  23....31....47....53....61....73....83
  89....359...389...401...449...479...491
  139...181...241...283...337...409...421
For example, 31 is in row 3 because 31+2*3 is a prime, unlike 31+2*1 and 31+2*2.  Every prime occurs exactly once.  For each row, it is not known whether it is finite.

Cf. A192176, A192177, A192178, A192179.

z = 5000; (* z=number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1,
      PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {_, True}];
Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[2 x + Prime[#1]]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 16}]; TableForm[Array[Prime[row[#]] &, {10}]] (* A192175 array *)
Flatten[Table[ Prime[row[k][[n - k + 1]]], {n, 1, 11}, {k, 1, n}]] (* A192175 sequence *)
(* Peter J. C. Moses, Jun 20 2011 *)

A238691 a(n) = A190339(n)/A224911(n).

Original entry on oeis.org

1, 2, 3, 15, 15, 21, 1155, 165, 2145, 51051, 255255, 440895, 440895, 969, 111435, 248834355, 248834355, 2927463, 5898837945, 44352165, 1641030105, 8563193457, 42815967285, 80047243185, 1360803134145, 32898537309, 7731156267615, 1028243783592795, 1028243783592795, 375840831244263

Offset: 0

Paul Curtz, Mar 03 2014

nonn

Are non-repeated terms of A224911(n) (2,3,5,11,17,...) A124588(n+1)?
Are repeated terms of A224911(n) (7,13,19,23,31,37,...) A049591(n+1)? At that sequence, Benoit Cloitre mentions a link to the Bernoulli numbers.
Greatest primes dividing a(n): 1, 2, 3, 5, 5, 7, 11, 11, 13, 17, 17, 19, 19, 19, 23, 29, 29, 29, ... = b(n). It appears that b(n) is A224911(n) with A008578(n), ancient primes, instead of A000040(n).
Hence c(n) = 2, 6, 15, 35, ... = 2, followed by A006094(n+1).

			a(0)=2/2=1, a(1)=6/3=2, a(2)=15/5=3, a(3)=a(4)=105/7=15, ... .

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A060308.

nmax = 40; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; (#/FactorInteger[#][[-1, 1]])& /@ Denominator[Diagonal[diff]]

a(16)-a(25) from Jean-François Alcover, Mar 03 2014

A152916 Tetrahedral numbers k(k+1)(k+2)/6 such that exactly two of k, k+1, and k+2 are prime.

Original entry on oeis.org

1, 4, 10, 35, 286, 969, 4495, 12341, 35990, 62196, 176851, 209934, 437989, 562475, 971970, 1179616, 1293699, 1975354, 2303960, 3280455, 3737581, 5061836, 7023974, 12347930, 13436856, 16435111, 23706021, 30865405, 35999900, 39338069

Offset: 1

Juri-Stepan Gerasimov, Dec 15 2008

nonn

			k=1: Of the three numbers (1,2,3), exactly two are prime, so 1*2*3/6 = 1 is in the sequence.
k=2: Of the three numbers (2,3,4), exactly two are prime, so 2*3*4/6 = 4 is in the sequence.
k=4: Of the three numbers (4,5,6), exactly one is prime, so 4*5*6/6 = 20 is not in the sequence.

Cf. A000040, A000292, A141468, A144521, A152622.

A000292 := proc(n) n*(n+1)*(n+2)/6; end: for n from 1 to 800 do ps := 0 ; if isprime(n) then ps := ps+1 ; fi; if isprime(n+1) then ps := ps+1 ; fi; if isprime(n+2) then ps := ps+1 ; fi; if ps = 2 then printf("%d,", A000292(n)) ; fi; od: # R. J. Mathar, Aug 14 2009

a(n) = A000292(A124588(n-1)), n > 1. - R. J. Mathar, Aug 14 2009

Name and Example section clarified by Jon E. Schoenfield, Aug 06 2017

A178378 Products of 2 primes that are the difference of 2 primes.

Original entry on oeis.org

4, 6, 9, 10, 15, 22, 25, 33, 34, 51, 55, 58, 82, 85, 87, 118, 121, 123, 142, 145, 177, 187, 202, 205, 213, 214, 274, 289, 295, 298, 303, 319, 321, 355, 358, 382, 394, 411, 447, 451, 454, 478, 493, 505, 535, 537, 538, 562, 573, 591, 622, 649, 681, 685, 694, 697

Offset: 1

Juri-Stepan Gerasimov, May 26 2010

nonn

			a(1)=4 because 4=2*2=(5-3)*(5-3) where 2,3,5 and 5-3 are all prime, a(2)=6 because 6=2*3=(5-3)*(5-2) where 2,3,5,5-3 and 5-2 are all prime.

Cf. A000040, A001358.

a(n)=A124588(m)*A124588(k).

Extended beyond 478 by R. J. Mathar, May 28 2010

A178656 The positions of products of 2 primes that are the differences of 2 primes in A002808.

Original entry on oeis.org

1, 2, 4, 5, 8, 13, 15, 21, 22, 35, 38, 41, 59, 61, 63, 87, 90, 92, 107, 110, 136, 144, 155, 158, 165, 166, 215, 227, 232, 235, 240, 252, 254, 283, 286, 306, 316, 330, 360, 363, 366, 386, 398, 408, 435, 437, 438, 459, 467, 483, 507, 530, 557, 560, 568, 571, 589

Offset: 1

Juri-Stepan Gerasimov, Jun 01 2010

nonn

The positions of A124588(m)*A124588(k) in A002808.

A002808(a(n))=A124588(m)*A124588(k).

Corrected (254 inserted) by R. J. Mathar, Jun 04 2010

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A029710 Primes such that next prime is 4 greater.

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A124589 Primes p such that q-p <= 4, where q is the next prime after p.

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A192175 Array of primes determined by distance to next prime, by antidiagonals.

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A238691 a(n) = A190339(n)/A224911(n).

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A152916 Tetrahedral numbers k*(k+1)*(k+2)/6 such that exactly two of k, k+1, and k+2 are prime.

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Maple

Formula

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A178378 Products of 2 primes that are the difference of 2 primes.

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Author

Keywords

Examples

Crossrefs

Formula

Extensions

A178656 The positions of products of 2 primes that are the differences of 2 primes in A002808.

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Author

Keywords

Comments

Formula

Extensions

A152916 Tetrahedral numbers k(k+1)(k+2)/6 such that exactly two of k, k+1, and k+2 are prime.