A091180 -id:A091180 - cp's OEIS frontend

A136064 Mother primes of order 5.

23, 67, 199, 331, 397, 463, 661, 727, 859, 1123, 1783, 2113, 2179, 2311, 2971, 3037, 3433, 3631, 3697, 4027, 4093, 4159, 4357, 4621, 5347, 5479, 5743, 6007, 6271, 6337, 6733, 7393, 7591, 7789, 8053, 8317, 8647, 9043, 9109, 9439, 9967, 10099, 10627

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180. For mother primes of order 2 see A136061. For mother primes of order 3 see A136062. For mother primes of order 4 see A136063.

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

Cf. A088878, A091180, A136019, A136020, A136061, A136062, A136063, A136065, A136066, A136067, A136068, A136069, A136070.

n = 5; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136065 Mother primes of order 6.

Original entry on oeis.org

53, 79, 131, 157, 521, 547, 599, 677, 859, 911, 937, 1249, 1301, 1327, 1951, 2029, 2237, 2341, 2549, 2731, 2887, 2939, 3121, 3251, 3329, 3407, 3511, 3797, 4057, 4759, 4967, 5591, 5981, 6007, 6761, 7229, 7307, 7411, 7489, 7879, 8009, 8191, 8581, 8737

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180. For mother primes of order 2 see A136061. For mother primes of order 3 see A136062. For mother primes of order 4 see A136063. For mother primes of order 5 see A136064.

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

Cf. A088878, A091180, A136019, A136020, A136061, A136062, A136063, A136064, A136066, A136067, A136068, A136069, A136070.

n = 6; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136067 Mother primes of order 8.

Original entry on oeis.org

103, 307, 613, 1021, 1123, 1327, 2143, 2347, 2551, 3061, 3571, 3877, 4591, 6427, 6733, 7753, 8263, 8467, 9181, 9283, 10303, 10711, 11731, 12037, 12343, 12547, 12853, 15607, 15913, 16831, 17137, 17341, 17851, 18973, 19891, 21013, 21727

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180. For mother primes of order 2 see A136061. For mother primes of order 3 see A136062. For mother primes of order 4 see A136063. For mother primes of order 5 see A136064. For mother primes of order 6 see A136065. For mother primes of order 8 see A136066.

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

Cf. A088878, A091180, A136019, A136020, A136061, A136062, A136063, A136064, A136065, A136066, A136068, A136069, A136070.

n = 8; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136068 Mother primes of order 9.

Original entry on oeis.org

191, 229, 419, 571, 761, 1103, 1483, 1559, 1901, 2053, 2129, 2851, 3079, 4219, 4409, 4523, 4561, 4751, 6271, 6689, 6803, 7069, 7753, 8171, 8209, 8513, 8741, 8779, 9311, 9463, 9539, 10831, 11743, 11971, 12161, 12503, 12541, 12959, 14251, 14593, 14669

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180. For mother primes of order 2 see A136061. For mother primes of order 3 see A136062. For mother primes of order 4 see A136063. For mother primes of order 5 see A136064. For mother primes of order 6 see A136065. For mother primes of order 8 see A136066. For mother primes of order 9 see A136067.

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

Cf. A088878, A091180, A136019, A136020, A136061, A136062, A136063, A136064, A136065, A136066, A136067, A136069, A136070.

n = 9; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A136070 Mother primes of order 11.

Original entry on oeis.org

47, 139, 277, 691, 829, 967, 1381, 1657, 2347, 3727, 4831, 5107, 5521, 6211, 7039, 7177, 7591, 8419, 9109, 9661, 10627, 12007, 12421, 13249, 14767, 16699, 17389, 19597, 20149, 20287, 21529, 24151, 24979, 25117, 26497, 28429, 29671, 29947

Offset: 1

Artur Jasinski, Dec 12 2007

nonn

For smallest mother primes of order n see A136020 (also definition). For mother primes of order 1 see A091180. For mother primes of order 2 see A136061. For mother primes of order 3 see A136062. For mother primes of order 4 see A136063. For mother primes of order 5 see A136064. For mother primes of order 6 see A136065. For mother primes of order 8 see A136066. For mother primes of order 9 see A136067. For mother primes of order 10 see A136068.

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

Cf. A088878, A091180, A136019, A136020, A136061, A136062, A136063, A136064, A136065, A136066, A136067, A136068, A136069.

n = 11; a = {}; Do[If[PrimeQ[(Prime[k] + 2n)/(2n + 1)], AppendTo[a, Prime[k]]], {k, 1, 1500}]; a

A091179 A088878 indexed by A000040.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 12, 14, 15, 16, 18, 19, 20, 27, 30, 31, 33, 38, 39, 42, 43, 44, 47, 54, 55, 56, 58, 59, 62, 63, 65, 67, 68, 69, 74, 79, 83, 84, 86, 89, 91, 93, 94, 99, 100, 102, 107, 108, 110, 122, 123, 129, 133, 134, 135, 139, 143, 147, 153, 154, 155, 156, 162, 167

Offset: 1

Ray Chandler, Dec 27 2003

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

Cf. A088878, A091180, A091181.

select(n -> isprime(3*ithprime(n)-2), [$1..1000]); # Robert Israel, Mar 04 2016

PrimePi@ Select[Prime@ Range@ 167, PrimeQ[3 # - 2] &] (* Michael De Vlieger, Mar 04 2016 *)

a(n)=k such that A000040(k) = A088878(n).

Offset corrected by Michael De Vlieger, Mar 04 2016

A265763 Numerators of primes-only best approximates (POBAs) to 3; see Comments.

Original entry on oeis.org

7, 5, 17, 13, 23, 19, 31, 41, 37, 53, 59, 71, 67, 89, 113, 109, 131, 127, 139, 157, 179, 181, 199, 211, 239, 251, 269, 293, 311, 307, 337, 383, 379, 409, 419, 449, 491, 487, 503, 499, 521, 541, 571, 577, 593, 599, 631, 683, 701, 719, 751, 773, 769, 787, 809

Offset: 1

Clark Kimberling, Dec 18 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, and also, |x - p/q| < |x - p'/q| for every prime p' except p. 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 A265759 for a guide to related sequences.

			The POBAs for 3 start with 7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

Cf. A000040, A023208, A088878, A091180, A094525, A222565.

x = 3; 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, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)

A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

Original entry on oeis.org

2, 2, 5, 5, 7, 7, 11, 13, 13, 17, 19, 23, 23, 29, 37, 37, 43, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 103, 113, 127, 127, 137, 139, 149, 163, 163, 167, 167, 173, 181, 191, 193, 197, 199, 211, 227, 233, 239, 251, 257, 257, 263, 269, 271, 277, 293

Offset: 1

Clark Kimberling, Dec 18 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, and also, |x - p/q| < |x - p'/q| for every prime p' except p. 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 A265759 for a guide to related sequences.

			The POBAs for 3 start with  7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

Cf. A000040, A023208, A088878, A091180, A094525, A265763.

x = 3; 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, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)

A338410 Primes p such that (p+2)/3 and (p+3)/2 are prime.

Original entry on oeis.org

7, 19, 31, 139, 199, 211, 379, 499, 631, 919, 1039, 1291, 1399, 1759, 2179, 2719, 2731, 2971, 3271, 3691, 4591, 5791, 5851, 6079, 7591, 8011, 8779, 10039, 11299, 11719, 11731, 12979, 14251, 15031, 15511, 15679, 18451, 18859, 20071, 21379, 21559, 22051, 22639, 23599, 24499, 24691, 25339, 25579

Offset: 1

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

nonn

All terms == 7 (mod 12).

			a(3) = 31 is in the sequence because 31, (31+2)/3 = 11 and ((31+3)/2) = 17 are prime.

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

Intersection of A091180 and A092109.

filter:= t -> isprime(t) and isprime((t+2)/3) and isprime((t+3)/2):
select(filter, [seq(i,i=7..30000,12)]);

Select[Prime[Range[3000]],AllTrue[{(#+2)/3,(#+3)/2},PrimeQ]&] (* Harvey P. Dale, May 20 2023 *)

isok(p) = iferr(isprime(p) && isprime((p+2)/3) && isprime((p+3)/2), E, 0); \\ Michel Marcus, Oct 25 2020

A257002 Primes p such that p+2 divides p^p+2.

Original entry on oeis.org

7, 13, 19, 31, 37, 61, 67, 109, 127, 139, 157, 181, 193, 199, 211, 307, 313, 337, 379, 397, 409, 487, 499, 541, 571, 577, 631, 691, 751, 769, 787, 811, 829, 877, 919, 937, 991, 1009, 1021, 1039, 1117, 1201, 1291, 1297, 1327, 1381, 1399, 1459, 1471, 1531, 1567

Offset: 1

K. D. Bajpai, Apr 14 2015

nonn

All the terms in this sequence are congruent to 1 mod 3.

Primes p such that 2^p == 2 (mod p+2). Includes A091180. - Robert Israel, Apr 14 2015

			a(1) = 7 is prime; 7+2 = 9 divides 7^7 + 2 = 823545.
a(2) = 13 is prime; 13+2 = 15 divides 13^13 + 2 = 302875106592255.

K. D. Bajpai, Table of n, a(n) for n = 1..5700

Cf. A000040, A091180, A213382.

[ p: p in PrimesUpTo(1600) | (p^p+2) mod (p+2) eq 0 ]; // Vincenzo Librandi, Apr 15 2015

select(t -> isprime(t) and (2 &^t - 2) mod (t+2) = 0, [seq(6*i+1,i=1..10^4)]); # Robert Israel, Apr 14 2015

Select[Prime[Range[3000]], Mod[#^# + 2, # + 2] == 0 &]
Select[Prime[Range[500]],PowerMod[#,#,#+2]==#&] (* Harvey P. Dale, May 19 2017 *)

forprime(p=2,1000, if(Mod(p^p+2,p+2)==0, print1(p, ", ")));

from sympy import prime
A257002_list = [p for p in (prime(n) for n in range(1,10**4)) if pow(p, p, p+2) == p] # Chai Wah Wu, Apr 14 2015

cp's OEIS Frontend

A136064 Mother primes of order 5.

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A136065 Mother primes of order 6.

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A136067 Mother primes of order 8.

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A136068 Mother primes of order 9.

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A136070 Mother primes of order 11.

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A091179 A088878 indexed by A000040.

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A265763 Numerators of primes-only best approximates (POBAs) to 3; see Comments.

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A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

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A338410 Primes p such that (p+2)/3 and (p+3)/2 are prime.

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