A007693 -id:A007693 - cp's OEIS frontend

A051651 Primes of the form 30030*p + 1 where p is a prime.

150151, 330331, 390391, 870871, 930931, 1231231, 1411411, 1831831, 2012011, 2372371, 2672671, 3813811, 4174171, 4474471, 5375371, 5435431, 5735731, 5915911, 8078071, 9219211, 10120111, 10420411, 11021011, 11501491, 12642631, 14024011, 14624611, 16246231, 16426411

Offset: 1

Labos Elemer

nonn

Generalization of A005385; can be called 30030-safe primes.

A002110(6)*p + 1 = 30030*p + 1 (prime).

			390391 is a term because 390391 = 2*3*5*7*11*13*p + 1 is prime, where p = 13.

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

Cf. A002110, A005384, A005385, A007693, A051654.

Select[30030Prime[Range[100]]+1,PrimeQ] (* Harvey P. Dale, Dec 09 2010 *)

isok(k) = isprime(k) && k % 30030 == 1 && isprime((k-1)/30030); \\ Amiram Eldar, Feb 24 2025

a(n) = 30030 * A051654(n) + 1. - Amiram Eldar, Feb 24 2025

A075704 p and 12*p+1 are both primes.

Original entry on oeis.org

3, 5, 13, 19, 23, 29, 31, 59, 61, 71, 73, 83, 89, 101, 103, 139, 149, 191, 199, 223, 229, 233, 269, 271, 281, 293, 311, 379, 383, 401, 409, 433, 463, 479, 503, 523, 569, 601, 631, 643, 661, 691, 719, 751, 761, 773, 811, 829, 839, 863, 883, 929, 953, 1009, 1013

Offset: 0

Jani Melik, Oct 02 2002

nonn

			5 is a prime and 12*5+1=61 is also a prime. 13 and 12*13+1=157 are both primes...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A005384, A023212, A007693, A023228, A023237.

ts_m_sophie_germain_pras := proc(n); if (isprime(n)='true' and isprime(12*n+1)='true') then RETURN(n); fi; end: seq(ts_m_sophie_germain_pras(i), i=1..2030);

Select[Prime[Range[300]],PrimeQ[12#+1]&] (* Harvey P. Dale, Feb 06 2012 *)

A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.

Original entry on oeis.org

13, 11, 19, 17, 31, 29, 43, 41, 67, 79, 103, 101, 113, 139, 137, 173, 223, 257, 283, 281, 317, 353, 367, 401, 439, 499, 607, 619, 617, 643, 641, 653, 677, 761, 787, 823, 821, 907, 941, 977, 1039, 1087, 1181, 1193, 1361, 1373, 1399, 1433, 1447, 1543, 1579

Offset: 1

Clark Kimberling, Dec 20 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 to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265771.

x = 6; 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, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)

A265771 Denominators of primes-only best approximates (POBAs) to 6; see Comments.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 7, 7, 11, 13, 17, 17, 19, 23, 23, 29, 37, 43, 47, 47, 53, 59, 61, 67, 73, 83, 101, 103, 103, 107, 107, 109, 113, 127, 131, 137, 137, 151, 157, 163, 173, 181, 197, 199, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 283, 283, 293, 311

Offset: 1

Clark Kimberling, Dec 20 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 to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265770.

x = 6; 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, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)

A188583 Primes p such that 6*p^3+1 is also prime.

Original entry on oeis.org

3, 5, 13, 41, 97, 131, 223, 283, 353, 397, 461, 467, 523, 577, 661, 677, 691, 773, 811, 887, 937, 997, 1091, 1223, 1277, 1321, 1447, 1487, 1567, 1571, 1637, 1721, 1741, 1777, 1823, 1861, 2161, 2243, 2273, 2341, 2351, 2357, 2371, 2383, 2467, 2551

Offset: 1

Bruno Berselli, Apr 22 2011

nonn

			For prime  p = 283,  6*p^3+1 = 135991123  is prime.

Bruno Berselli, Table of n, a(n) for n = 1..900

Cf. A052291, A005384; A007693, A153812.

[ p: p in PrimesUpTo(2600) | IsPrime(6*p^3+1) ];

Select[Prime[Range[500]],PrimeQ[6#^3+1]&] (* Harvey P. Dale, May 13 2011 *)

A374593 Numbers k such that k - rad(k) + 1 is prime, where rad(k) is the radical A007947(k).

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 32, 36, 40, 44, 45, 48, 49, 50, 56, 60, 63, 72, 75, 80, 81, 84, 88, 90, 92, 99, 104, 108, 116, 117, 128, 132, 136, 140, 144, 147, 153, 156, 160, 162, 164, 168, 169, 180, 184, 200, 204, 207, 212, 216, 224, 225, 234, 240, 243, 245, 250

Offset: 1

Arsen Vardanyan, Aug 23 2024

nonn

Includes 4*p for p in A005384, 8*p for p in A007693, and 16*p for p in A228857. - Robert Israel, Jun 27 2025

			12 is a term, because 12 - rad(12) + 1 = 12 - (2*3) + 1 = 12 - 6 + 1 = 7 is prime.

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

Cf. A005384, A007693, A007947, A097379, A228857, A359213.

rad:= n -> convert(numtheory:-factorset(n),`*`):
select(k -> isprime(k - rad(k)+1), [$1..1000]); # Robert Israel, Jun 27 2025

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);Select[Range[250],PrimeQ[#-rad[#]+1]&] (* James C. McMahon, Sep 27 2024 *)

isok(k) = isprime(k - (factorback(factor(k)[, 1])) + 1);

A075705 Safe primes (A005385) (p and (p-1)/2 are primes) such that 6*p+1 is also prime.

Original entry on oeis.org

5, 7, 11, 23, 47, 83, 107, 263, 347, 467, 503, 863, 887, 1283, 1487, 1823, 1907, 2027, 2063, 2447, 2903, 3203, 3623, 4007, 4127, 4547, 4703, 4787, 5387, 5807, 7523, 7703, 8147, 8423, 11423, 11483, 11807, 12107, 12227, 12647, 12983, 13043, 13163, 14087, 14207

Offset: 1

Jani Melik, Oct 02 2002

nonn

			47 is prime, so is (47-1)/2=23 and also 6*47+1=283; 83 is a prime, (83-1)/2=41 and 6*83+1=499, ...

David Consiglio, Jr. and T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A059455, A007693.

ts_sg_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sg_var_pras(i), i=1..3000);

Select[Range[20000], PrimeQ[#] && PrimeQ[(#-1)/2] && PrimeQ[6#+1] &] (* T. D. Noe, Nov 07 2011 *)
Select[Prime[Range[1700]],And@@PrimeQ[{(#-1)/2,6#+1}]&] (* Harvey P. Dale, Feb 28 2013 *)

A106059 Primes p such that p + 6 and 6*p + 1 are primes.

Original entry on oeis.org

5, 7, 11, 13, 17, 23, 37, 47, 61, 73, 83, 101, 103, 107, 131, 151, 173, 233, 257, 263, 271, 277, 311, 331, 347, 367, 373, 443, 461, 503, 557, 593, 601, 607, 641, 653, 727, 751, 853, 941, 947, 971, 1013, 1033, 1063, 1091, 1103, 1117, 1283, 1321, 1361

Offset: 1

Zak Seidov, May 07 2005

nonn

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

Cf. A007693, A023201.

[p: p in PrimesUpTo(5000)|IsPrime(p+6) and IsPrime(6*p+1)]; // Vincenzo Librandi, Jan 30 2011

Select[Prime[Range[220]], PrimeQ[6#+1]&&PrimeQ[1#+6]&]

A153591 Primes p such that 6p^2+6p+1 is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 23, 31, 41, 43, 47, 67, 73, 97, 101, 107, 109, 137, 151, 199, 233, 239, 241, 251, 263, 283, 317, 331, 337, 353, 359, 379, 383, 419, 421, 431, 439, 449, 463, 541, 569, 571, 577, 601, 619, 647, 691, 701, 769, 823, 839, 877, 907, 929, 953, 971

Offset: 1

Vincenzo Librandi, Dec 29 2008

			For p = 2, 6p^2+6p+1 = 37 is prime; for p = 47, 6p^2+6p+1 = 13537 is prime.

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

Cf. A007693 (n and 6n+1 are primes), A153812 (primes p such that 6*p^2+1 is also prime).

[ p: p in PrimesUpTo(1000) | IsPrime(6*p^2+6*p+1) ];

Select[Prime[Range[200]], PrimeQ[6#^2 + 6# + 1] &] (* Vincenzo Librandi, Oct 15 2012 *)

Edited and extended beyond a(20) by Klaus Brockhaus, Jan 01 2009

A188132 Primes p such that p == 3 (mod 4) and 6p+1 is prime.

Original entry on oeis.org

3, 7, 11, 23, 47, 83, 103, 107, 131, 151, 263, 271, 283, 311, 331, 347, 367, 443, 467, 503, 607, 683, 727, 751, 787, 863, 887, 907, 947, 971, 1063, 1091, 1103, 1151, 1171, 1283, 1327, 1423, 1427, 1451, 1487, 1511, 1531, 1567, 1607, 1787, 1811, 1823, 1831, 1847, 1907, 1931, 1987

Offset: 1

M. F. Hasler, Mar 21 2011

nonn

Complement of A188131 in A007693 \ {2}.

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

Cf. A002515, A007693, A188131.

Select[Range[3, 2000, 4], PrimeQ[#] && PrimeQ[6# + 1] &] (* Amiram Eldar, Nov 13 2019 *)

forprime( p=1,1e4, p%4==3 & isprime(p*6+1) & print1(p", "))

cp's OEIS Frontend

A051651 Primes of the form 30030*p + 1 where p is a prime.

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A075704 p and 12*p+1 are both primes.

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Maple

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A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.

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Mathematica

A265771 Denominators of primes-only best approximates (POBAs) to 6; see Comments.

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Mathematica

A188583 Primes p such that 6*p^3+1 is also prime.

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Magma

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A374593 Numbers k such that k - rad(k) + 1 is prime, where rad(k) is the radical A007947(k).

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Maple

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PARI

A075705 Safe primes (A005385) (p and (p-1)/2 are primes) such that 6*p+1 is also prime.

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Maple

Mathematica

A106059 Primes p such that p + 6 and 6*p + 1 are primes.

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