cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A178421 Lower primes p1 in a twin pair such that sum of p1 and p2 yields average a1 of twin prime pairs and product of 2*a1 is another average of twin prime pairs.

Original entry on oeis.org

211049, 248639, 253679, 410339, 507359, 605639, 1121189, 1138829, 1262099, 2162579, 2172869, 2277659, 4070219, 6305459, 7671509, 11659409, 12577109, 14203769, 14862119, 17472839, 18728639, 18798359, 20520569, 21140699
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 27 2010

Keywords

Comments

The definition means that a1/2, a1 and 2*a1 are all in A014574 (twin prime averages). - R. J. Mathar, Nov 02 2023

Examples

			211049 is a term since 211049 and 211051 are twin primes; 211049 + 211051 = 422100 is an average of twin primes, and 2*422100 = 844200 is another average of twin primes.

Links

Crossrefs

Cf. A069175, A014574, A069142, A099349.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=Prime[n+1];a1=p1+p2;a2=2*a1;If[p2-p1==2&&PrimeQ[a1-1]&&PrimeQ[a1+1]&&PrimeQ[a2-1]&&PrimeQ[a2+1],AppendTo[lst,p1]],{n,10!}];lst
atpQ[{a_,b_}]:=Module[{m=a+b},b-a==2&&AllTrue[m+{1,-1},PrimeQ] && AllTrue[ 2m+{1,-1},PrimeQ]]; Select[Partition[Prime[Range[134*10^4]],2,1],atpQ][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2019 *)

Formula

a(n) = A069175(n)-1. - R. J. Mathar, Nov 02 2023

A176821 List of 4-tuples of twin primes q, p, p+2 and q+2 such that 2*q

Original entry on oeis.org

5, 11, 13, 7, 29, 59, 61, 31, 659, 1319, 1321, 661, 809, 1619, 1621, 811, 2129, 4259, 4261, 2131, 2549, 5099, 5101, 2551, 3329, 6659, 6661, 3331, 3389, 6779, 6781, 3391, 5849, 11699, 11701, 5851, 6269, 12539, 12541, 6271, 10529, 21059, 21061, 10531
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 26 2010, May 01 2010, May 07 2010

Keywords

Comments

The first number q in each quadruplet is in A069142 (equivalent to selecting twin primes q which are also Sophie-Germain primes). [From R. J. Mathar, May 06 2010]

Crossrefs

Cf. A077800, A045536.

Extensions

Corrected (2131 replaced by 3331) by R. J. Mathar, May 06 2010

A329151 Primes p such that (3*p+q)/2 and (p+3*q)/2 are prime, where q is the next prime after p.

Original entry on oeis.org

5, 29, 47, 139, 293, 557, 607, 647, 659, 709, 773, 809, 811, 863, 1117, 1171, 1187, 1801, 2129, 2467, 2477, 2549, 2917, 3023, 3229, 3329, 3389, 3469, 3593, 3617, 3833, 4261, 4363, 5791, 5849, 6269, 6781, 6949, 7069, 7703, 8273, 9397, 9973, 10141, 10343, 10369, 10529, 10657, 11059, 11329, 13063
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Feb 25 2020

Keywords

Comments

Intersection with A001359 is A069142.

Examples

			a(3)=47 is a term because it is prime, the next prime is 53, and (3*47+53)/2 = 97 and (47+3*53)/2 = 103 are both prime.

Links

Crossrefs

Cf. A001359, A069142.

Programs

  • Maple
    Primes:= [seq(ithprime(i),i=1..10000)]:
Res:= NULL:
for i from 2 to nops(Primes)-1 do
   p:= Primes[i]; q:= Primes[i+1]; c:= (p+q)/2;
   if isprime(p+c) and isprime(q+c) then
     Res:= Res, p
   fi
od:
Res;

A336300 Primes p such that there exists k with p < k < q such that p+k and q+k are prime, where q is the next prime after p.

Original entry on oeis.org

5, 23, 29, 31, 47, 53, 73, 83, 113, 131, 139, 151, 173, 199, 211, 251, 293, 317, 359, 373, 467, 491, 509, 523, 541, 557, 593, 607, 647, 659, 683, 709, 719, 761, 773, 797, 809, 811, 863, 911, 991, 997, 1063, 1069, 1103, 1117, 1163, 1171, 1187, 1201, 1231, 1259, 1307, 1327, 1409, 1439, 1459, 1499
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Oct 22 2020

Keywords

Comments

There may be more than one such k: thus for a(17) = 293, k = 294, 300, and 306 all work.

Examples

			a(4)=31 is included because 31 is prime, the next prime after 31 is 37, and with k=36 we have 31+36=67 and 37+36=73 both prime.

Links

Crossrefs

Includes A069142.

Programs

  • Maple
    P:= [seq(ithprime(i),i=2..1000)]:
R:= NULL:
for i from 1 to nops(P)-1 do
  for k from ceil(P[i]/2) to floor(P[i+1]/2) do
    if isprime(P[i]+2*k) and isprime(P[i+1]+2*k) then
      R:= R, P[i]; break
    fi
  od;
od:
R;

A068635 Smaller of the first twin pair to yield a twin pair chain of length n.

Original entry on oeis.org

3, 5, 211049, 253679, 41887255409, 73768891456259
Offset: 1

Views

Author

Neil Fernandez, Apr 10 2002

Keywords

Comments

PARI program written by Rick L. Shepherd.
If n is in the sequence and n>5 then Mod[n,30]=29. - Farideh Firoozbakht, Mar 26 2004

Examples

			5,7 form a twin pair, separated by the number 6. Their sum 12 is the separator of another twin pair, 11,13. _Their_ sum is not the separator of a further twin pair. (5,7),(11,13) is a chain of length 2. 5 is the lower of the smallest twin pair to generate a chain of this length, so a(2)=5.
Here are the actual chains (lesser of twin primes given): 3; 5 -> 11; 211049 -> 422099 -> 844199; 253679 -> 507359 -> 1014719 -> 2029439; 41887255409 -> 83774510819 -> 167549021639 -> 335098043279 -> 670196086559; ...

References

Crossrefs

Cf. A069142. Equals A069179 - 1.

Programs

  • PARI
    for(n=1,40000,if(prime(n+1)-prime(n)==2, s=prime(n+1)+prime(n); if(isprime(s-1)&&isprime(s+1), if(isprime(2*s-1)&&isprime(2*s+1),print(n,", ", prime(n))))))

A261463 Numbers n such that n is a twin prime and 2n + 1 is a twin prime.

Original entry on oeis.org

3, 5, 29, 659, 809, 2129, 2549, 3329, 3389, 5849, 6269, 10529, 33179, 41609, 44129, 53549, 55439, 57329, 63839, 65099, 70379, 70979, 72269, 74099, 74759, 78779, 80669, 81929, 87539, 93239, 102299, 115469, 124769, 133979, 136949, 156419, 161459, 168449
Offset: 1

Views

Author

Harvey P. Dale, Aug 20 2015

Keywords

Comments

n is a Sophie Germain prime and a twin prime, and 2*n+1 is also a twin prime.
Apparently this contains 3 and the members of A069142. - R. J. Mathar, Aug 23 2015

Examples

			809 is a term because 809 and 811 are twin primes, and 2*809+1 = 1619 is a prime and a twin prime with 1621.

Links

Programs

  • Mathematica
    sgtpQ[n_]:=Module[{sgp=2n+1},PrimeQ[sgp]&&AnyTrue[sgp+{2,-2},PrimeQ]]; Select[Union[Flatten[Select[Partition[Prime[Range[25000]],2,1], #[[2]]- #[[1]] ==2&]]],sgtpQ] (* The program uses the AnyTrue function from Mathematica version 10 *)
p=PrimeQ; Select[Prime@ Range[10^5], p[2#+1] && (p[#+2] || p[#-2]) && (p[2#+3] || p[2#-1]) &] (* Giovanni Resta, Aug 20 2015 *)

Formula

a(n+1) = A069142(n), n>=1. - G. C. Greubel, Aug 23 2015

A328036 Even semiprimes such that the next semiprime is also even.

Original entry on oeis.org

4, 10, 58, 274, 382, 454, 458, 538, 614, 1082, 1234, 1318, 1478, 1618, 1718, 1874, 2062, 2374, 2554, 2846, 2902, 3574, 3722, 3998, 4174, 4258, 4474, 4946, 5098, 5414, 5422, 5498, 6334, 6598, 6658, 6686, 6718, 6778, 6914, 6922, 7054, 7346, 7534, 7642, 7702, 7754, 7838, 8038, 8098, 8422, 8434, 8674, 8818
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Oct 02 2019

Keywords

Examples

			a(3)=58 is a member because 58=2*29 is an even semiprime and the next semiprime is 62, which is also even.

Links

Crossrefs

Cf. A001358. Includes 2*A069142.

Programs

  • Maple
    sp:= select(t -> numtheory:-bigomega(t)=2, [$1..20000]):
sp[select(t -> sp[t]::even and sp[t+1]::even, [$1..nops(sp)-1])];
  • Mathematica
    Select[Partition[Select[Range[10000],PrimeOmega[#]==2&],2,1],AllTrue[#,EvenQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 01 2020 *)
Showing 1-7 of 7 results.

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