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 1-9 of 9 results.

A174922 Lesser of twin primes p1 such that p1+(p2^2-p1^2) and p2+(p2^2-p1^2) are prime numbers.

Original entry on oeis.org

5, 11, 29, 461, 599, 659, 809, 1019, 1289, 2027, 2141, 2309, 2339, 2801, 3329, 3557, 3581, 4127, 4421, 4547, 5879, 6761, 10091, 10457, 10709, 13829, 15329, 18911, 20231, 21839, 23561, 23909, 26249, 26879, 27581, 27689, 27917, 28109, 30491
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5+(7^2-5^2)=5+24=29; 7+(7^2-5^2)=7+24=31,..

Links

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A174920

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1+(p2^2-p1^2)]&&PrimeQ[p2+(p2^2-p1^2)],AppendTo[lst,p1]],{n,8!}];lst
prQ[{a_,b_}]:=Module[{c=b^2-a^2},AllTrue[{a+c,b+c},PrimeQ]]; Transpose[ Select[ Select[ Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]] == 2&], prQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 12 2015 *)

A132929 Averages of twin primes such that the sum of the lower, average and upper parts of the twin primes are averages of other twin primes.

Original entry on oeis.org

4, 6, 60, 270, 1950, 3000, 6360, 11490, 11550, 14550, 18540, 19890, 21840, 31080, 32910, 32970, 33330, 33600, 42570, 42840, 50460, 53550, 58110, 68880, 70200, 74610, 79230, 80910, 93810, 96330, 98910, 104310, 109140, 114600, 121020, 125790
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 26 2008

Keywords

Examples

			4 is a term since (3, 5) are twin primes, 3 + 4 + 5 = 12 and (11, 13) are also twin primes.
6 is a term since (5, 7) are twin primes, 5 + 6 + 7 = 18 and (17, 19) are also twin primes.

Links

Crossrefs

Cf. A014574.
Cf. A174920, A177336.

Programs

  • Mathematica
    TwinPrimeAverageQ[n_]:=If[PrimeQ[n-1]&&PrimeQ[n+1],True,False](*TwinPrimeAverageQ*) lst={};Do[If[TwinPrimeAverageQ[n],If[TwinPrimeAverageQ[3*n],(*Print[n];*)AppendTo[lst,n]]],{n,9!}];lst

Formula

From Wesley Ivan Hurt, May 03 2022: (Start)
a(n) = A174920(n) + 1.
a(n) = A177336(n) - 1. (End)

A177335 List of 4-tuples of twin primes q, p, p+2 and q+2 such that 3*q

Original entry on oeis.org

3, 11, 13, 5, 5, 17, 19, 7, 59, 179, 181, 61, 269, 809, 811, 271, 1949, 5849, 5851, 1951, 2999, 8999, 9001, 3001, 6359, 19079, 19081, 6361, 11489, 34469, 34471, 11491, 11549, 34649, 34651, 11551, 14549, 43649, 43651, 14551
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 08 2010

Keywords

Links

Crossrefs

Cf. A174920, A176821.

Programs

  • Maple
    A174920:= select(q -> isprime(q) and isprime(q+2) and isprime(3*q+2) and isprime(3*q+4), [3,seq(i,i=5..200000,6)]):
map(t -> (t, 3*t+2, 3*t+4, t+2), A174920); # Robert Israel, May 05 2019

Formula

From Robert Israel, May 05 2019: (Start)
a(4k-3) = A174920(k).
a(4k-2) = 3*A174920(k) + 2.
a(4k-1) = 3*A174920(k) + 4.
a(4k) = A174920(k)+2. (End)

Extensions

Verified and extended by D. S. McNeil, May 10 2010

A177336 Greater of twin primes p such that 3*p-2 is also greater of twin primes.

Original entry on oeis.org

5, 7, 61, 271, 1951, 3001, 6361, 11491, 11551, 14551, 18541, 19891, 21841, 31081, 32911, 32971, 33331, 33601, 42571, 42841, 50461, 53551, 58111, 68881, 70201, 74611, 79231, 80911, 93811, 96331, 98911, 104311, 109141, 114601, 121021, 125791
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 07 2010

Keywords

Examples

			a(1) = 5 because 5 is the greater of the twin primes (3, 5) and 3*5 - 2 = 13 is the greater of the twin primes (11, 13).

Links

Crossrefs

Cf. A006512, A038869, A088878.
Cf. A132929, A174920.

Programs

  • Magma
    [p:p in PrimesInInterval(3,130000)| IsPrime(p-2) and IsPrime(3*p-2) and IsPrime(3*p-4)]; // Marius A. Burtea, Dec 23 2019
  • Mathematica
    Select[Range[3, 126000], And @@ PrimeQ[{#, # - 2, 3# - 2, 3# - 4}] &] (* Amiram Eldar, Dec 23 2019 *)

Formula

From Wesley Ivan Hurt, May 03 2022: (Start)
a(n) = A132929(n) + 1.
a(n) = A174920(n) + 2. (End)

Extensions

Definition corrected, 1231 and 1483 inserted, and all values above 3000 corrected by R. J. Mathar, May 10 2010
Terms corrected to match definition by D. S. McNeil, May 10 2010
Name corrected by Amiram Eldar, Dec 23 2019

A174955 Lesser of twin primes p1 such that p1*p2+-6 are prime numbers.

Original entry on oeis.org

5, 11, 1061, 2111, 3371, 3851, 5867, 9461, 12251, 21491, 22037, 22481, 24917, 26681, 28277, 32141, 42641, 43607, 48731, 56477, 59417, 59627, 67271, 67757, 70487, 77417, 86531, 87221, 91127, 104147, 113621, 115151, 116687, 119291, 121577
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5*7=35+-6 -> primes,..

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A174920, A174922

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2+6]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst

A242772 The lesser of twin primes p1 such that 2*p1 + p2 is a prime number (A174913) and also the lesser of other twin primes in A174913.

Original entry on oeis.org

5, 11489, 32969, 33329, 33599, 42839, 58109, 93809, 96329, 114599, 180179, 272999, 309539, 334889, 401309, 540539, 633569, 717089, 784349, 820409, 870239, 879689, 907139, 948089, 989249, 991619, 994559, 1020959, 1028579, 1044749, 1185659, 1189649, 1245449, 1253909
Offset: 1

Views

Author

Ivan N. Ianakiev, May 22 2014

Keywords

Comments

It seems that a(n) == 9 mod 10 for n > 1.
a(n) == 9 (mod 10) for n > 1 since if p1 == 1, 3 or 7 (mod 10) then 2*p1 + p2, p2, or 2*p1 + p2 + 2 is divisible by 5, respectively. - Amiram Eldar, Dec 31 2019

Examples

			a(1) = A174913(2) = 5 and 2*5 + 7 = 17 = A174913(3).

Links

Crossrefs

Cf. A001359, A174913, A174920, A242773.

Programs

  • Mathematica
    Select[Range[10^6], And @@ PrimeQ[{#, # + 2,(p = 3*# + 2), p + 2, 3*p + 2}] &] (* Amiram Eldar, Dec 31 2019 *)
  • PARI
    isok(p) = isprime(p) && isprime(p+2) && isprime(q=3*p+2) && isprime(q+2) && isprime(3*q+2); \\ Michel Marcus, May 23 2014

A352132 Numbers k such that k, k+4, 3*k+4 and 3*k+8 are all semiprimes.

Original entry on oeis.org

6, 10, 118, 119, 129, 155, 287, 295, 299, 319, 377, 413, 447, 469, 511, 538, 629, 681, 699, 717, 785, 831, 865, 913, 1003, 1073, 1077, 1111, 1115, 1137, 1141, 1145, 1267, 1343, 1345, 1379, 1393, 1437, 1469, 1509, 1687, 1817, 1835, 1919, 1923, 1981, 2167, 2173, 2177, 2195, 2245, 2429, 2479, 2569
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Mar 05 2022

Keywords

Comments

Numbers k such that k and 3*k+4 are both in A175648.
Even terms are 2*k for k in A174920.

Examples

			a(4) = 119 is a term because 119 = 7*17, 119+4 = 123 = 3*41, 3*119+4 = 361 = 19^2 and 3*119+8 = 365 = 5*73 are semiprimes.

Links

Crossrefs

Cf. A001358, A174920, A175648.

Programs

  • Maple
    filter:= proc(x)
numtheory:-bigomega(x) = 2 and numtheory:-bigomega(x+4) = 2 and numtheory:-bigomega(3*x+4) = 2 and numtheory:-bigomega(3*x+8)=2
end proc:
select(filter, [$1..3000]);
  • Mathematica
    okQ[k_] := AllTrue[{k, k+4, 3k+4, 3k+8}, PrimeOmega[#] == 2&];
Select[Range[3000], okQ] (* Jean-François Alcover, May 16 2023 *)

A174957 Lesser of twin primes p1 such that p1*p2-4 and p1*p2-6 are twin prime numbers.

Original entry on oeis.org

5, 11, 1031, 2711, 3851, 4421, 5867, 8837, 10067, 12041, 12251, 12611, 17957, 21491, 21521, 22037, 22481, 23537, 32141, 32411, 42641, 48311, 48731, 49367, 50261, 53231, 60167, 72167, 77417, 80147, 80447, 81047, 87641, 88337, 90527, 95231
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Apr 02 2010

Keywords

Comments

5*7=35; 35-4=31; 35-6=29; 29,31 twin primes

Links

Crossrefs

Cf. A001359, A174913, A174915, A174916, A174917, A174920, A174922, A174955.

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2-4]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst
ltp[{a_,b_}]:=b-a==2&&AllTrue[a*b-{4,6},PrimeQ]; Select[Partition[Prime[ Range[ 10000]],2,1],ltp][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 07 2017 *)

A327700 Primes p such that p + q*(q-p) and q + p*(q-p) are prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 23, 59, 61, 83, 151, 233, 263, 269, 293, 373, 401, 433, 503, 541, 619, 701, 971, 1103, 1433, 1493, 1601, 1621, 1861, 1949, 2099, 2179, 2371, 2441, 2543, 2741, 2851, 2903, 2999, 3083, 3181, 3313, 3413, 3559, 3631, 4073, 4093, 4549, 4591, 4643, 5039, 5081, 5471, 5711, 5749
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Sep 22 2019

Keywords

Links

Crossrefs

Includes A174920.

Programs

  • Maple
    R:= NULL: count:= 0:
q:= 2:
do
  p:= q; q:= nextprime(p);
  if isprime(p+(q-p)*q) and isprime(q+(q-p)*p) then
     count:= count+1;
     R:= R, p;
     if count = 100 then break fi
  fi
od:
R;
  • Mathematica
    Do[a=Prime[k]+Prime[k+1]*(Prime[k+1]-Prime[k]);b=Prime[k+1]+Prime[k]*(Prime[k+1]-Prime[k]);If[PrimeQ[a]&&PrimeQ[b],Print[Prime[k]]],{k,1,757}] (* Metin Sariyar, Sep 23 2019 *)
chpQ[{a_,b_}]:=AllTrue[{a+b(b-a),b+a(b-a)},PrimeQ]; Select[Partition[ Prime[ Range[800]],2,1],chpQ][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2021 *)
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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