A098058 -id:A098058 - cp's OEIS frontend

A339415 Table read by rows. If p=A098058(n+1), q is the next prime after p, and r=(p+q)/2, row n consists of the areas (in increasing order) of triangles with vertices (p,p), (s,r-s), (q,q), where s and r-s are prime.

0, 0, 2, 4, 8, 0, 36, 60, 4, 8, 16, 0, 36, 72, 84, 4, 16, 20, 32, 36, 72, 108, 132, 54, 90, 150, 2, 14, 22, 26, 34, 46, 54, 90, 126, 162, 174, 10, 14, 34, 46, 50, 62, 54, 90, 126, 198, 210, 0, 144, 180, 216, 240, 16, 20, 40, 44, 56, 64, 76, 92, 14, 26, 34, 50, 70, 86, 94, 98, 14, 98, 182, 266

Offset: 1

J. M. Bergot and Robert Israel, Dec 03 2020

If p = A098058(n+1), r is an even number >=4, and Goldbach's conjecture implies that r is the sum of primes s and r-s.
By symmetry, s and r-s produce the same area; only one of these is included in the table.
The row includes 0 if and only if r/2 is prime, i.e. p is in A339414.

			With p=A098058(5)=17, q=19, r=18, the values of s are 5, 7, 11, 13, corresponding to areas 4, 8, 8, 4 respectively, so row 4 is (4,8).
The first 10 rows are
0
0
2
4, 8
0, 36, 60
4, 8, 16
0, 36, 72, 84
4, 16, 20, 32
36, 72, 108, 132
54, 90, 150

Robert Israel, Table of n, a(n) for n = 1..10020 (rows 1 to 261, flattened)

Cf. A098058, A339414.

R:= 0: count:= 1: q:= 5: nrows:= 1:
printf("0\n"):
while nrows < 20  do
  p:= q; q:= nextprime(q);
  if p+q mod 4 <> 0 then next fi;
  nrows:= nrows+1;
  r:= (p+q)/2;
  T:= select(t -> isprime(t) and isprime(r-t), [$ceil(r/2)..r]);
  count:= count + nops(T);
  V:= map(t -> abs((p-q)*(p+q-4*t)/4), T);
  R:= R, op(V);
  printf("%a\n",V);
od:

The area of the triangle with vertices (p,p), (s,r-s), (q,q) is (q-p)*|p+q-4*s|/4.

A259311 First differences of A098058.

Original entry on oeis.org

1, 2, 6, 6, 6, 6, 2, 10, 6, 6, 6, 2, 10, 2, 10, 18, 6, 6, 18, 6, 2, 10, 2, 6, 10, 6, 6, 2, 10, 6, 30, 6, 6, 2, 10, 6, 6, 6, 2, 10, 2, 10, 18, 6, 14, 6, 10, 6, 14, 6, 10, 26, 10, 2, 10, 2, 10, 18, 42, 18, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 10

Offset: 1

Zak Seidov, Jun 24 2015

nonn

No term is divisible by 4, see comments in A098058.

Cf. A098058.

lista(n) = for(x=1, n, z=(prime(x+1)-prime(x)); if(z%4, print1(z, ", "))) \\ Michel Marcus, Jun 24 2015

a(n) = A098058(n+1)-A098058(n).

A154939 Primes p such that (p-1)*(p+1)-+p are primes.

Original entry on oeis.org

3, 5, 11, 31, 101, 131, 149, 181, 241, 331, 419, 449, 709, 1051, 1061, 1171, 1409, 1549, 1579, 1699, 1759, 1831, 2069, 3229, 3449, 3761, 3911, 4159, 4951, 5821, 6029, 6481, 6661, 6679, 6899, 7079, 7151, 7229, 7369, 8101, 8219, 8629, 8861, 9091, 9161, 9521

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

That is, primes p such that p^2+p-1 and p^2-p-1 are both primes: intersection of A053184 and A091567. - Michel Marcus, Jul 10 2016

			2*4=8-+3 -> primes, 4*6=24-+5 -> primes,...

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

Cf. A053184, A038872, A141158, A091567, A098058, A038936, A089270, A140559.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p-1) and IsPrime(p^2-p-1)]; // Vincenzo Librandi, Jul 10 2016

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*(p+1)-p]&&PrimeQ[(p-1)*(p+1)+p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]], And@@PrimeQ/@{#^2 - # - 1, #^2 + # - 1} &] (* Vincenzo Librandi, Jul 10 2016 *)
Select[Prime[Range[1500]],AllTrue[(#-1)(#+1)+{#,-#},PrimeQ]&] (* Harvey P. Dale, Sep 21 2023 *)

A098059 Primes preceding gaps divisible by 4.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 89, 97, 103, 109, 127, 163, 193, 199, 211, 223, 229, 277, 307, 313, 349, 359, 379, 389, 397, 401, 439, 449, 457, 463, 467, 479, 487, 491, 499, 509, 613, 619, 643, 661, 673, 683, 701, 719, 739, 743, 757, 761, 769, 797, 823, 853, 859

Offset: 1

Cino Hilliard, Sep 11 2004

Prime complement of A098058. - Robert G. Wilson v, Jul 17 2015

			7 is a term since the next prime after 7 is 11 and 11-7 is divisible by 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A152087.

Cf. A001223, A098058.

N:= 1000:  # to get all terms up to the second-last prime <= N
Primes:= select(isprime,[2, 2*i+1 $ i=1..floor((N-1)/2)]):
Gaps:= Primes[2..-1] - Primes[1..-2]:
Primes[select(t-> Gaps[t] mod 4 = 0, [$1..nops(Gaps)])]; # Robert Israel, Jun 24 2015

Prime[Select[Range[150], Mod[Prime[ # + 1] - Prime[ # ], 4] == 0 &]] (* Ray Chandler, Oct 26 2006 *)
Transpose[Select[Partition[Prime[Range[200]],2,1],Divisible[Last[#]- First[#], 4]&]][[1]] (* Harvey P. Dale, Apr 06 2013 *)

f(n) = for(x=1,n,z=(prime(x+1)-prime(x));if(z%4==0,print1(prime(x)",")))

p=2; forprime(q=3,1e4, if((q-p)%4==0, print1(p", ")); p=q) \\ Charles R Greathouse IV, Jun 29 2015

a(n) ~ 2n log n. - Charles R Greathouse IV, Jun 29 2015

Edited by Ray Chandler, Oct 26 2006

New name from Robert Israel and Charles R Greathouse IV, Jun 29 2015

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

Original entry on oeis.org

5, 7, 13, 23, 37, 43, 73, 167, 233, 263, 433, 557, 587, 593, 607, 727, 857, 1153, 1597, 1627, 1753, 2143, 2663, 2713, 3433, 3607, 3863, 3947, 4027, 4363, 4423, 4673, 5147, 5477, 5623, 5807, 5903, 6277, 7237, 7333, 7577, 8287, 8647, 8837, 8887, 9043, 10067

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*7-10=11, 3*7+10=31,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939

lst={};Do[p=Prime[n];If[PrimeQ[(p-2)*(p+2)-2*p]&&PrimeQ[(p-2)*(p+2)+2*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-2)(#+2)+{2#,-2#},PrimeQ]&] (* Harvey P. Dale, Jan 01 2025 *)

A253969 Primes p such that p + nextprime(p) is divisible by 6.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 59, 67, 71, 79, 89, 97, 101, 103, 107, 109, 113, 127, 137, 139, 149, 163, 179, 181, 191, 193, 197, 223, 227, 229, 239, 241, 269, 277, 281, 283, 293, 307, 311, 313, 317, 337, 347, 349, 359, 379, 389, 397, 401, 409, 419, 421

Offset: 1

Vincenzo Librandi, Jan 21 2015

Sequence provides all primes p such that p + nextprime(p) is divisible by 3 (see Crossrefs).

			p=19 is in this sequence because 19+23 = 42 is divisible by 6.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. sum of prime p and next prime after p is divisible by k: A000040 (k=2, after the first term), this sequence (k=3, k=6), A098058(k=4, after the first term), A253970 (k=5; k=10 without 2), A179849 (k=7), A253972 (k=8), A253973 (k=9).

[p: p in PrimesUpTo(500) | (p+NextPrime(p)) mod 6 eq 0];

Prime@Select[Range[200], Mod[Prime[#] + Prime[# + 1], 6]==0 &]
Select[Partition[Prime[Range[100]],2,1],Divisible[Total[#],6]&][[All,1]] (* Harvey P. Dale, Jan 20 2018 *)

forprime(p=1,10^3,if(!((p+nextprime(p+1))%6),print1(p,", "))) \\ Derek Orr, Jan 30 2015

A289118 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 4k+1 and 4k+3 or 4k+3 and 4k+1.

Original entry on oeis.org

3, 3, 3, 23, 47, 131, 131, 233, 233, 521, 521, 521, 521, 521, 521, 51749, 505049, 1391087, 2264839, 2556713, 2569529, 2569529, 6160043, 6160043, 6160043, 43679609, 43679609, 198572029, 701575297, 5552898499, 6639843979, 9005520203, 9005520203, 99052377023

Offset: 1

Jonathan Sondow, Jun 25 2017

nonn

Conjecture: the sequence is infinite. (Motivation: the string HTHTHT. . of length n eventually occurs in any sufficiently long sequence of coin tosses.)

			{Prime[k], Mod[ Prime[k], 4]} = {{3, 3}, {5, 1}, {7, 3}, {11, 3}, {13, 1}, {17, 1}, {19, 3}, {23, 3}, {29, 1}}, {31, 3}, {37, 1}, . . for k = 2, 3, 4, . ., so a(n) = 3, 3, 3, 23 for n = 1, 2, 3, 4.

R. K. Guy, Unsolved Problems in Number Theory, A4.

Giovanni Resta, Table of n, a(n) for n = 1..45
Jens Kruse Andersen, Consecutive Congruent Primes

For the least prime at the start of such a string of length exactly n, see A247384.

Cf. A098058, A289119.

j = 2; T = Table[ While[ Product[ Mod[ Prime[k + 1] - Prime[k], 4], {k, j, j + n}] == 0,  j++]; Prime[j], {n, 0, 15}]; Prepend[T, 3]

a(n) = A247384(n) if and only if n > 1 and a(n) < a(n+1).

a(18)-a(27) from Alois P. Heinz, Jun 26 2017

a(28)-a(34) from Giovanni Resta, Jul 02 2017

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

Original entry on oeis.org

7, 17, 37, 113, 157, 227, 283, 293, 313, 347, 443, 587, 787, 883, 1063, 1097, 1237, 1303, 1327, 1427, 1567, 1723, 1933, 1973, 2087, 2347, 2467, 2687, 2777, 3457, 3593, 4447, 4703, 4793, 4967, 5737, 5827, 6317, 6607, 6793, 6857, 8297, 8563, 8803, 9433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

4*10-3*7=19, 4*10+3*7=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006

lst={};Do[p=Prime[n];If[PrimeQ[(p-3)*(p+3)-3*p]&&PrimeQ[(p-3)*(p+3)+3*p],AppendTo[lst,p]],{n,7!}];lst

A247384 Find the first (maximal) string of consecutive primes of length exactly n which alternate between 4k+1 and 4k+3 or 4k+3 and 4k+1 as in A002144(4n+1) and A002145(4n+3). The first element is a(n).

Original entry on oeis.org

97, 11, 3, 23, 47, 167, 131, 2011, 233, 23633, 34499, 1013, 9341, 90659, 521, 51749, 505049, 1391087, 2264839, 2556713, 17123893, 2569529, 15090641, 18246451, 6160043, 1557431471, 43679609, 198572029, 701575297, 5552898499, 6639843979, 61233611783, 9005520203

Offset: 1

J. M. Bergot, Sep 15 2014

nonn

			a(4)=23 because 23,29,31,37 alternate 4*n+3,4*n+1,4*n+3,4*n+1 for exactly four primes and 23 is the least prime for a string of exactly four.

Jens Kruse Andersen and Giovanni Resta, Table of n, a(n) for n = 1..45 (first 38 terms from Jens Kruse Andersen)
Jens Kruse Andersen, Consecutive Congruent Primes

Cf. A002144, A002145, A098058, A098059, A289118, A289237.

Primes:= select(isprime,[seq(2*i+1,i=1..10^7)]):
Pm4:= map(`modp`,[seq((-1)^j*Primes[j],j=1..nops(Primes))],4):
Starts:= [1,op(select(t -> Pm4[t-1]<> Pm4[t], [$2..nops(Pm4)]))]:
Lengths:= [seq(Starts[i+1]-Starts[i],i=1..nops(Starts)-1)]:
for i from 1 to max(Lengths) do A[i]:= ListTools:-Search(i,Lengths) od:
R:=[seq(A[i],i=1..max(Lengths))]:
seq(`if`(a=0,0,Primes[Starts[a]]),a=R); # Robert Israel, Sep 15 2014

i = 2; While[ Mod[ Prime[i] - Prime[i - 1], 4] != 0 || Mod[ Prime[i + 1] - Prime[i], 4] != 0, i++]; T = {Prime[i]}; Do[j = 2; While[! (Product[ Mod[ Prime[k + 1] - Prime[k], 4], {k, j, j + n}] != 0 && (Mod[Prime[j] - Prime[j - 1], 4] == 0 || j == 2) && Mod[ Prime[j + n + 2] - Prime[j + n + 1], 4] == 0), j++]; T = Append[T, Prime[j]], {n, 0, 13}]; T (* Jonathan Sondow, Jun 28 2017 *)

v=vector(100);v[1]=7;cur=1;p=3;forprime(q=5, 1e10, if((q-p)%4==0,if(!v[cur],v[cur]=back(p,cur);print("a("cur") = "v[cur]));cur=1,cur++);p=q) \\ Charles R Greathouse IV, Sep 15 2014

a(n) = A289118(n) if and only if n > 1 and A289118(n) < A289118(n+1). - Jonathan Sondow, Jun 27 2017

More terms from Jens Kruse Andersen, Oct 01 2014

Definition clarified by Jonathan Sondow, Jun 25 2017

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.

Original entry on oeis.org

3, 5, 7, 11, 19, 29, 31, 59, 101, 139, 239, 271, 829, 1031, 1201, 1439, 1511, 1531, 2251, 2609, 3929, 4349, 4969, 5449, 5639, 5711, 5801, 5881, 5981, 6521, 6569, 6701, 6949, 6959, 8221, 8831, 9001, 9181, 9209, 9419, 9511, 9929, 10139, 10711, 11839, 11981

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 18 2009

nonn

3*11-28=5, 3*11+28=61, ...

Cf. A125272, A053184, A038872, A141158, A038615, A098058, A038936, A089270, A140559, A154939, A155006, A155007

lst={};Do[p=Prime[n];If[PrimeQ[(p-4)*(p+4)-4*p]&&PrimeQ[(p-4)*(p+4)+4*p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]],AllTrue[(#-4)(#+4)+{4#,-4#},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2020 *)

cp's OEIS Frontend

A339415 Table read by rows. If p=A098058(n+1), q is the next prime after p, and r=(p+q)/2, row n consists of the areas (in increasing order) of triangles with vertices (p,p), (s,r-s), (q,q), where s and r-s are prime.

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A259311 First differences of A098058.

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PARI

Formula

A154939 Primes p such that (p-1)*(p+1)-+p are primes.

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Magma

Mathematica

A098059 Primes preceding gaps divisible by 4.

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Maple

Mathematica

PARI

PARI

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A155006 Primes p such that (p-2)*(p+2)-+2*p are primes.

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Crossrefs

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Mathematica

A253969 Primes p such that p + nextprime(p) is divisible by 6.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A289118 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 4*k+1 and 4*k+3 or 4*k+3 and 4*k+1.

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Author

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Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A155006 Primes p such that (p-2)(p+2)-+2p are primes.

A289118 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 4k+1 and 4k+3 or 4k+3 and 4k+1.

A155007 Primes p such that (p-3)(p+3)-+3p are primes.

A247384 Find the first (maximal) string of consecutive primes of length exactly n which alternate between 4k+1 and 4k+3 or 4k+3 and 4k+1 as in A002144(4n+1) and A002145(4n+3). The first element is a(n).

A155008 Primes p such that (p-a)(p+a)-+ap are primes,a=4.