A007693 -id:A007693 - cp's OEIS frontend

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617

Offset: 1

Clark Kimberling, Dec 15 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x        Lower POBA       Upper POBA       POBA
1        A001359/A006512  A006512/A001359  A265759/A265760
3/2      A104163/A158708  A162336/A158709  A265761/A222565
2        A005383/A005382  A005385/A005384  A079149/A120628
3        A091180/A088878  A094525/A023208  A265763/A265764
4        A162857/A062737  A090866/A023212  A265765/A120639
5        A265766/A158318  A265767/A023217  A265768/A265769
6        A227756/A158015  A051644/A007693  A265770/A265771
sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
tau      A265796/A265797  A265798/A265799  A265800/A265801
1/tau    A265799/A265798  A265797/A265796  A265806/A265807
pi       A265808/A265809  A265810/A265811  A265812/A265813
e        A265814/A265815  A265816/A265817  A265818/A265819

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Cf. A000040, A001359, A006512, A265759, A265760.

x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A051644 Primes of the form 6*p + 1 where p is also prime.

Original entry on oeis.org

13, 19, 31, 43, 67, 79, 103, 139, 223, 283, 367, 439, 499, 607, 619, 643, 787, 823, 907, 1039, 1087, 1399, 1447, 1543, 1579, 1627, 1663, 1699, 1759, 1867, 1879, 1987, 2083, 2203, 2239, 2383, 2659, 2767, 2803, 3019, 3343, 3463, 3559, 3607, 3643, 3847, 3919

Offset: 1

Labos Elemer

Analogous to A005385; can be called 6-safe primes.

			103 is in the sequence because both 17 and 6*17 + 1 = 103 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A005384, A005385, A002476, A007693, A016921, A024899, A091178.

Select[1 + 6Prime@Range[120], PrimeQ] (* Ray Chandler, Mar 14 2007 *)

isok(k) = isprime(k) && k % 6 == 1 && isprime((k-1)/6); \\ Amiram Eldar, Feb 24 2025

a(n) = 6*A007693(n) + 1.

Edited, corrected and extended by Ray Chandler, Mar 14 2007

A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341

Offset: 1

David W. Wilson

nonn

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.

Entries in chains are congruent to {1,7,3,9} mod 10.

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766.

Subsequence of A007693, A023256, and A024899.

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010

k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Additional comments from Labos Elemer, Jul 23 2003

A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

Original entry on oeis.org

2, 2, 2, 2, 2, 5, 17, 11, 11, 11, 2, 23, 7, 43, 19, 3, 5, 2, 7, 3, 61, 53, 2, 41, 47, 2, 5, 7, 31, 2, 47, 13, 113, 7, 137, 103, 43, 41, 97, 3, 173, 97, 41, 13, 97, 59, 29, 53, 3, 107, 127, 197, 3, 487, 433, 31, 281, 587, 7, 89, 41, 47, 193, 239, 41, 7, 31, 67

Offset: 1

Labos Elemer, Dec 15 1999

nonn

a(n) is the minimal prime p such that primorial(n)*p + 1 is also prime.
While p is in A005384, the primorial(n)*p + 1 primes are in A051902 (primorial-safe primes).
Analogous to or subset of A051686, where the even numbers are 2, 6, ..., A002110(n), ...

			a(25) = 47 because primorial(25)*47 + 1 is also prime and minimal with this property: primorial(25)*47 + 1 = 47*2305567963945518424753102147331756070 + 1 = 108361694305439365963395800924592535291 is a minimal prime. The first 6 terms (2,2,2,2,2,5) correspond to first entries in A005384, A007693, A051645, A051647, A051653, A051654, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A002110, A005384, A005385, A051686, A007693, A051886, A051888, A051902.

Table[p = 2; While[! PrimeQ[Product[Prime@ i, {i, n}] p + 1], p = NextPrime@ p]; p, {n, 68}] (* Michael De Vlieger, Jun 29 2017 *)

a(n) = {my(p = 2, r = vecprod(primes(n))); while(!isprime(p * r + 1), p = nextprime(p+1)); p;} \\ Amiram Eldar, Feb 25 2025

a(n) = (A051902(n)-1)/A002110(n). - Amiram Eldar, Feb 25 2025

More terms from Michael De Vlieger, Jun 29 2017

A023256 Primes that remain prime through 2 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

2, 17, 47, 61, 101, 131, 151, 331, 367, 461, 607, 751, 937, 971, 1151, 1321, 1327, 1361, 1481, 1511, 1607, 1811, 1847, 1907, 2081, 2287, 2347, 2357, 2791, 2851, 2971, 3251, 3257, 3457, 3491, 3581, 3761, 4007, 4091, 4127, 4861, 4967, 5231, 5347, 5387, 5407

Offset: 1

David W. Wilson

nonn

Primes p such that 6*p+1 and 36*p+7 are also primes. - Vincenzo Librandi, Aug 04 2010

Subsequence of A007693. - Michel Marcus, Oct 17 2015

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A007693 (1 iteration), A023287 (3 iterations).

[n: n in [0..100000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7)] // Vincenzo Librandi, Aug 04 2010

Select[Prime[Range[800]],AllTrue[Rest[NestList[6#+1&,#,2]],PrimeQ]&] (* Harvey P. Dale, Oct 17 2020 *)

lista(nn) = forprime(n=1, nn, if (isprime(p=6*n+1) && isprime(6*p+1), print1(n, ", "))); \\ Michel Marcus, Oct 17 2015

a(n) == 1 or 7 (mod 10) for n > 1. - John Cerkan, Sep 14 2016

A051647 Primes p such that 210*p + 1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 47, 53, 59, 67, 73, 83, 89, 101, 137, 139, 157, 163, 179, 181, 191, 193, 223, 229, 251, 271, 277, 281, 313, 317, 347, 349, 353, 359, 401, 419, 421, 431, 433, 449, 457, 463, 479, 523, 577, 599, 601, 631, 653, 701, 709, 719, 727

Offset: 1

Labos Elemer

Analogous to A005384, Sophie Germain primes.

A002110(4)*p + 1 = 210*p + 1 and p are both primes.

			11 is in the sequence because 210*11 + 1 = 2311 is also prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A005384, A005385, A007693, A051648.

[p: p in PrimesUpTo(900) | IsPrime(210*p+1)]; // Vincenzo Librandi, Apr 11 2013

Select[Prime[Range[200]],PrimeQ[210#+1]&]  (* Harvey P. Dale, Apr 25 2011 *)

isok(k) = isprime(k) && isprime(210*k+1); \\ Amiram Eldar, Feb 24 2025

a(n) = (A051648(n)-1)/210. - Amiram Eldar, Feb 24 2025

A051645 Primes p such that 30*p+1 is also prime.

Original entry on oeis.org

2, 5, 7, 11, 19, 23, 41, 43, 61, 67, 71, 79, 89, 109, 113, 131, 137, 167, 181, 193, 223, 229, 233, 277, 281, 313, 317, 331, 337, 359, 383, 439, 443, 457, 461, 467, 491, 503, 509, 541, 547, 593, 599, 607, 641, 691, 701, 733, 739, 743, 751, 769, 797, 821, 823

Offset: 1

Labos Elemer

A002110(3)*p+1 and p are primes.

			p = 19 is in the sequence because 30*p+1 = 571 is also prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A002110, A005384, A005385, A007693, A051646.

[p: p in PrimesUpTo(900) | IsPrime(30*p+1)]; // Vincenzo Librandi, Apr 11 2013

Select[Prime[Range[900]], PrimeQ[30 # + 1] &] (* Vincenzo Librandi, Apr 11 2013 *)

isok(k) = isprime(k) && isprime(30*k+1); \\ Amiram Eldar, Feb 24 2025

a(n) = (A051646(n)-1)/30. - Amiram Eldar, Feb 24 2025

A051646 Primes of the form 30*p + 1 where p is also prime.

Original entry on oeis.org

61, 151, 211, 331, 571, 691, 1231, 1291, 1831, 2011, 2131, 2371, 2671, 3271, 3391, 3931, 4111, 5011, 5431, 5791, 6691, 6871, 6991, 8311, 8431, 9391, 9511, 9931, 10111, 10771, 11491, 13171, 13291, 13711, 13831, 14011, 14731, 15091, 15271, 16231, 16411, 17791, 17971

Offset: 1

Labos Elemer

Analogous to A005385, safe primes. Can be called 30-safe primes.

			61 is in the sequence because both 2 and 30*2 + 1 = 61 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005384, A005385, A007693, A051645, A128470.

Select[30 * Prime[Range[120]] + 1, PrimeQ] (* Amiram Eldar, Feb 24 2025 *)

isok(k) = isprime(k) && k % 30 == 1 && isprime((k-1)/30); \\ Amiram Eldar, Feb 24 2025

a(n) = A128470(A051645(n)) = 30 * A051645(n) + 1. - Amiram Eldar, Feb 24 2025

A051648 Primes of form 210*p + 1 where p is a prime.

Original entry on oeis.org

421, 631, 1051, 1471, 2311, 2731, 3571, 4831, 6091, 9871, 11131, 12391, 14071, 15331, 17431, 18691, 21211, 28771, 29191, 32971, 34231, 37591, 38011, 40111, 40531, 46831, 48091, 52711, 56911, 58171, 59011, 65731, 66571, 72871, 73291, 74131, 75391, 84211, 87991

Offset: 1

Labos Elemer

Generalization of A005385. Can be called 210-safe primes.

A002110(4)*p + 1 = 210*p + 1 (prime).

			631 is in the sequence because 631 = 210*p + 1, where p=3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005384, A005385, A007693, A002110.

Select[Table[210n+1,{n,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Dec 25 2016 *)

isok(k) = isprime(k) && k % 210 == 1 && isprime((k-1)/210); \\ Amiram Eldar, Feb 24 2025

a(n) = 210 * A051647(n) + 1. - Amiram Eldar, Feb 24 2025

A051649 Primes of the form 2310*p + 1 where p is a prime.

Original entry on oeis.org

4621, 11551, 25411, 43891, 108571, 164011, 168631, 224071, 251791, 261031, 316471, 321091, 348811, 376531, 385771, 459691, 528991, 552091, 607531, 626011, 718411, 723031, 732271, 764611, 801571, 815431, 875491, 995611, 1000231, 1055671, 1064911, 1106491, 1161931

Offset: 1

Labos Elemer

nonn

Generalization of A005385; can be called 2310-safe primes.

A002110(5)*p + 1 = 2310*p + 1 (prime).

			11551 is a term because 11551 = 2310*p + 1 is prime, where p = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A002110, A005384, A005385, A007693, A051653.

Select[Table[2310n+1,{n,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Jan 18 2017 *)

isok(k) = isprime(k) && k % 2310 == 1 && isprime((k-1)/2310); \\ Amiram Eldar, Feb 24 2025

a(n) = 2310 * A051653(n) + 1. - Amiram Eldar, Feb 24 2025

cp's OEIS Frontend

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

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A051644 Primes of the form 6*p + 1 where p is also prime.

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A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

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A051887 Minimal and special 2k-Germain primes, where 2k is in A002110 (primorial numbers).

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A023256 Primes that remain prime through 2 iterations of function f(x) = 6x + 1.

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A051647 Primes p such that 210*p + 1 is also prime.

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A051645 Primes p such that 30*p+1 is also prime.

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