A001359 -id:A001359 - cp's OEIS frontend

A324669 a(n) is the least k>0 such that A001359(n)+k^2 is in A001359.

6, 114, 162, 210, 24, 330, 6, 6, 18, 12, 30, 210, 6, 18, 120, 150, 330, 24, 6, 42, 30, 66, 96, 210, 180, 210, 42, 54, 60, 360, 6, 18, 630, 60, 210, 24, 30, 66, 24, 126, 30, 48, 1380, 24, 90, 102, 6, 30, 42, 18, 90, 90, 42, 54, 12, 36, 60, 186, 210, 12, 72, 24, 42, 24, 330, 60, 12

Offset: 2

J. M. Bergot and Robert Israel, Sep 03 2019

nonn

Offset is 2 because 3+k^2 is never in A001359.
All terms are divisible by 6.
The generalized Bunyakovsky conjecture implies that a(n) always exists, for n >= 2.
a(n) = 6 if and only if A001359(n) is in A248367.

			a(3) = 114 because A001359(3)=11, 11+114^2=13007 is in A001359, and no smaller k works.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A001359, A248367.

P:= select(isprime, {seq(i,i=3..10000,2)}):
TP:= sort(convert(P intersect map(`-`,P,2),list)):
f:= proc(p) local k;
for k from 6 by 6 do if isprime(p + k^2) and isprime(p + k^2 + 2) then return k fi od
end proc:
map(f, TP[2..-1]);

With[{s = Select[Prime@ Range[3, 332], PrimeQ[# + 2] &]}, Array[Block[{k = 1}, While[! AllTrue[s[[#]] + k^2 + {0, 2}, PrimeQ], k++]; k] &, Length@ s]] (* Michael De Vlieger, Sep 03 2019 *)

A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

Original entry on oeis.org

11, 29, 59, 641, 101, 347, 2309, 569, 1931, 521, 1787, 419, 1229, 1871, 3671, 2237, 6551, 1427, 21491, 1607, 12377, 4931, 1019, 23201, 809, 19697, 12539, 2549, 38921, 10709, 37547, 8819, 9239, 34031, 6089, 80447, 15581, 46049, 36341, 14867, 38237, 36779, 87509, 71261, 15137, 40427, 13679, 54917, 41141, 50891

Offset: 1

Artur Jasinski, Jan 12 2021

nonn

Lesser twin primes (with the exception of prime 3) are congruent to 5 modulo 6, which implies that distances between successive pairs of twin primes are 6*k.

			a(1)=11 because 11 - 5 = 6*1.
a(2)=41 because 41 - 29 = 6*2.
a(3)=59 because 59 - 41 = 6*3.

Martin Raab, Table of n, a(n) for n = 1..4508

Cf. A001359, A001359, A053319, A007530, A052380, A052381, A113274, A113275, A052350.

Table[a[n] = 0, {n, 1, 10000}]; Table[
b[n] = 0, {n, 1, 10000}]; qq = {}; prev = 5; Do[
If[Prime[n + 1] - Prime[n] == 2, k = (Prime[n] - prev)/6;
  If[b[k] == 0, a[k] = Prime[n]; b[k] = 1]; prev = Prime[n]], {n, 5,
  10000}]; list = Table[a[n], {n, 1, 50}]
(* Second program: *)
pp = Select[Prime[Range[10^4]], PrimeQ[#+2]&];
dd = Differences[pp];
a[n_] := pp[[FirstPosition[dd, 6n][[1]]+1]];
Array[a, 50] (* Jean-François Alcover, Jan 13 2021 *)

a(n) = A052350(n) + 6*n.

A350247 Positive integers k such that the concatenation of k and 11 is the lesser of a pair of twin primes (i.e., a term of A001359).

Original entry on oeis.org

3, 21, 27, 72, 90, 126, 183, 189, 192, 210, 216, 261, 267, 300, 315, 324, 342, 345, 360, 378, 387, 414, 477, 483, 540, 567, 633, 672, 681, 687, 714, 717, 744, 750, 777, 798, 828, 861, 870, 888, 918, 939, 987, 1011, 1029, 1038, 1080, 1182, 1260, 1266, 1281

Offset: 1

Lorenzo Sauras Altuzarra, Dec 21 2021

Every term is a multiple of 3.

Numbers k such that 100*k+11 and 100*k+13 are prime. - Chai Wah Wu, Jan 20 2022

			311, 2111, 2711, 7211, and 9011 are terms of A001359.

Cf. A001359, A350246.

terms := proc(n)
   local k, p, L:
   k, L := 0, []:
   while numelems(L) < n do
      k := k+1:
      p := parse(cat(k, 11)):
      if isprime(p) and isprime(p+2) then L := [op(L), k]: fi: od:
   L: end:

Select[Range[1282], AllTrue[# + {0, 2}, PrimeQ] &[100 # + 11] &] (* Michael De Vlieger, Dec 21 2021 *)

from itertools import count, islice
from sympy import isprime
def A350247_gen(startvalue=3): # generator of terms >= startvalue
    for n in count(max(3,startvalue+(3-startvalue%3)%3),3):
        if isprime(100*n+11) and isprime(100*n+13):
            yield n
A350247_list = list(islice(A350247_gen(),20)) # Chai Wah Wu, Jan 20 2022

A359638 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n prime factors, counted with multiplicity.

Original entry on oeis.org

601, 1429, 81547, 248749, 27140749, 310314157, 3566181247

Offset: 3

Hugo Pfoertner, Jan 16 2023

Cf. A001222, A001359, A062502, A359636, A359637.

a359638(maxp) = {my (k=3, pp=3); forprime (p=5, maxp, my (mi=oo, ma=0); if (p-pp>2, for (j=pp+1, p-1, my(mo=bigomega(j)); if(mo

A359641 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, all with exponent 1.

Original entry on oeis.org

307, 8929, 992263, 229658167, 28674536239

Offset: 2

Hugo Pfoertner, Jan 17 2023

			a(3) = 8929: 8930 = 2*5*19*47, 8931 = 3*13*229, 8932 = 2^2*7*11*29;
a(6) = 28674536239: a(6)+1 = 2^4*5*7*31*43*107*359, a(6)+2 = 3*13*23*151*269*787, a(6)+3 = 2*11*17*19*37*191*571.

Cf. A001222, A001359, A087436, A285800, A359640.

obi(x,m=0) = {my (x2=x>>valuation(x,2), o=omega(x2)); if (o2, for (j=pp+1, p-1, my (mo=obi(j)); if (mo

A362941 Numbers of the form (p+1)*(p+3) where (p,p+2) is a twin prime pair (cf. A001359).

Original entry on oeis.org

24, 48, 168, 360, 960, 1848, 3720, 5328, 10608, 11880, 19320, 22800, 32760, 37248, 39600, 52440, 58080, 73440, 80088, 97968, 121800, 177240, 187488, 214368, 273528, 326040, 361200, 383160, 413448, 436920, 657720, 677328, 687240, 737880, 779688, 1042440, 1067088, 1104600

Offset: 1

N. J. A. Sloane, Sep 10 2023, following a suggestion from Jean-Claude Babois

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001359, A108604.

((# + 1)*(# + 3)) & /@ Select[Prime[Range[200]], PrimeQ[# + 2] &] (* Amiram Eldar, Sep 10 2023 *)

a(n) = A108604(n) - 1. - Amiram Eldar, Sep 10 2023

A371896 a(n) is the length of the uninterrupted sequence of primes generated by the polynomial f(x) = x^2 + x + p for x=0,1,..., where p=A001359(n).

Original entry on oeis.org

2, 4, 10, 16, 2, 40, 2, 2, 4, 3, 2, 2, 2, 3, 2, 4, 2, 2, 2, 3, 5, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 3, 2, 3, 3, 2, 2, 2, 2, 4, 2, 2, 7, 2, 3, 2, 5, 2, 4, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Peter Rowlett, Apr 11 2024

nonn

p=A001359(n) is the smaller prime of a twin prime pair so that f(0) = p and f(1) = p+2 are both primes so a(n) >= 2 and this sequence is the terms >= 2 in A208936.

			For n=6, p = A001359(n) = 41 and f(x) = x^2 + x + 41 is Euler's polynomial which generates primes f(x) for x=0,1,2,...,39, which is 40 terms so a(6) = 40 (cf. A202018).

L. Euler, Nouveaux Mémoires de l'Académie royale des Sciences, 1772, p. 36.

Peter Rowlett, Table of n, a(n) for n = 1..10000

Cf. A001359, A202018, A208936.

A001359[1] = 3;
A001359[n_] := A001359[n] = (p = NextPrime[A001359[n - 1]];
  While[!PrimeQ[p + 2], p = NextPrime[p]]; p)
A371896[n_] := With[{p = A001359@n}, k = 1;
  While[PrimeQ[k^2 + k + p], k++]; k]
(* Leo C. Stein, May 09 2024 *)

A375775 For n >= 1, a(n) is the largest k >= 1 such that A001359(n) + i*(i + 1) is prime for all i from 1 to k.

Original entry on oeis.org

1, 3, 9, 15, 1, 39, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 4, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 6, 1, 2, 1, 4, 1, 3, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Ctibor O. Zizka, Aug 27 2024

nonn

Record values of k for n = 1,2,3,4,6, i.e., for primes 3,5,11,17,41. What is the next record value of k if it exists ?
From Robert Israel, Sep 27 2024: (Start)
Dickson's conjecture implies that the sequence should be unbounded.  However, terms > 39 are expected to be extremely rare.  For 7 <= n <= 3*10^6 the only term > 9 is a(740969) = 10. (End)

			n = 1: A001359(1) = 3, 3 + 2 = 5, 3 + 6 is not a prime, thus k = 1.
n = 2: A001359(2) = 5, 5 + 2 = 7, 5 + 6 = 11, 5 + 12 = 17, 5 + 20 is not a prime, thus k = 3.

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

Cf. A000040, A001359, A002378.

Primes:= select(isprime, {seq(i,i=3..4000,2)}):
Twins:= Primes intersect map(`-`,Primes,2):
f:= proc(n) local i;
     for i from 1 do if not isprime(n+i*(i+1)) then return i-1 fi od
end proc:
map(f, Twins); # Robert Israel, Sep 27 2024

s[n_] := If[PrimeQ[n] && PrimeQ[n + 2], Module[{i = 1}, While[PrimeQ[n + i*(i + 1)], i++]; i - 1], Nothing]; Array[s, 3500] (* Amiram Eldar, Aug 27 2024 *)

\\ uses A001359 PARI code
a(n) = my(p=A001359(n)); for (k=1, oo, for (i=1, k, if (!isprime(p+i*(i + 1)), return(k-1)))); \\ Michel Marcus, Aug 27 2024

f(p) = for (k=1, oo, for (i=1, k, if (!isprime(p+i*(i + 1)), return(k-1))));
lista(nn) = my(v = select(x->isprime(x+2), primes(nn))); apply(f, v); \\ Michel Marcus, Aug 27 2024

A093326 Duplicate of A001359.

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487

Offset: 1

Giovanni Teofilatto, Apr 25 2004

dead

Name was: Primes p such that (p+p')/2+1 is a prime, where p' is the next prime after p.

lista(nn) = {forprime(p=3, nn, if (isprime(1+(p + nextprime(p+1))\2), print1(p, ", ")););} \\ Michel Marcus, Jun 03 2013

After correction a duplicate of A001359, Michel Marcus, Jun 03 2013

A097972 Least m such that both p|m and p+2|m+2 for twin prime pairs (p,p+2) (p=A001359).

Original entry on oeis.org

18, 40, 154, 340, 928, 1804, 3658, 5254, 10504, 11770, 19180, 22648, 32578, 37054, 39400, 52210, 57838, 73168, 79804, 97654, 121450, 176818, 187054, 213904, 273004, 325468, 360598, 382540, 412804, 436258, 656908, 676504, 686410, 737020, 778804

Offset: 1

Lekraj Beedassy, Sep 07 2004

nonn

			For instance, (a(4), a(4)+2), i.e., (340=17*20, 342=19*18) is the smallest pair whose elements are respectively divisible by the 4th twin prime pair (17, 19).

lm[{a_,b_}]:=Module[{k=a+1},While[!Divisible[k+2,b]||!Divisible[k,a],k++];k]; lm/@Select[Partition[Prime[Range[200]],2,1],#[[2]]- #[[1]] == 2&] (* Harvey P. Dale, Jun 06 2021 *)

a(n)=p*(p+3), with p=A001359.

cp's OEIS Frontend

A324669 a(n) is the least k>0 such that A001359(n)+k^2 is in A001359.

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A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

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A350247 Positive integers k such that the concatenation of k and 11 is the lesser of a pair of twin primes (i.e., a term of A001359).

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A359638 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n prime factors, counted with multiplicity.

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A359641 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, all with exponent 1.

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A362941 Numbers of the form (p+1)*(p+3) where (p,p+2) is a twin prime pair (cf. A001359).

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A371896 a(n) is the length of the uninterrupted sequence of primes generated by the polynomial f(x) = x^2 + x + p for x=0,1,..., where p=A001359(n).

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A375775 For n >= 1, a(n) is the largest k >= 1 such that A001359(n) + i*(i + 1) is prime for all i from 1 to k.

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