A023212 -id:A023212 - cp's OEIS frontend

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

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 *)

A087634 Primes p such that the equation phi(x) = 4p has a solution, where phi is the totient function.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 37, 41, 43, 53, 67, 73, 79, 83, 89, 97, 113, 127, 131, 139, 163, 173, 179, 191, 193, 199, 233, 239, 251, 277, 281, 293, 307, 359, 373, 409, 419, 431, 433, 443, 487, 491, 499, 509, 577, 593, 619, 641, 653, 659, 673, 683, 709, 719, 727

Offset: 1

T. D. Noe, Oct 24 2003

nonn

Except for p=2, the complement of A043297. Note that for primes p < 1000, we need to check for solutions x < 18478. The equation phi(x) = 2p has solutions for Sophie Germain primes, A005384

a(n) is also the primes p with 2p+1 or 4p+1 also prime, sequences A005384 and A023212. For the case 2p+1 a trivial solution is phi(6p+3)=4p, and for 4p+1, phi(4p+1)=4p. - Enrique Pérez Herrero, Aug 16 2011

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Totient Function

Cf. A005384, A043297.

Cf. A023212.

t=Table[EulerPhi[n], {n, 3, 20000}]; Union[Select[t, Mod[ #, 4]==0&&PrimeQ[ #/4]&& #/4<1000&]/4] (* or *)
Select[Prime[Range[100]],PrimeQ[4#+1]||PrimeQ[2#+1]&] (* Enrique Pérez Herrero, Aug 16 2011 *)

A182434 Number of primes p < n such that 4*p+1 is also prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Enrique Pérez Herrero, Apr 28 2012

nonn

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A023212, A182265.

Cf. A005384, A156660, A156874, A092816.

Accumulate[Table[Boole[PrimeQ[n]&&PrimeQ[4n+1]],{n,1,200}]]
Accumulate[If[AllTrue[{#,4#+1},PrimeQ],1,0]&/@Range[90]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 26 2015 *)

a(10^n) = A182265(n).

a(n) = sum(i=2..n, floor(phi(4*i^2+i)/(4*i^2-4*i))). - Enrique Pérez Herrero, May 02 2012.

A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

Original entry on oeis.org

11, 7, 13, 11, 19, 29, 43, 53, 67, 149, 163, 173, 211, 269, 283, 293, 317, 331, 389, 509, 523, 547, 557, 653, 691, 773, 787, 797, 907, 1051, 1109, 1123, 1171, 1229, 1493, 1531, 1637, 1723, 1733, 1867, 1949, 1997, 2011, 2083, 2251, 2309, 2347, 2371, 2467

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 4 start with  11/2, 7/2, 13/3, 11/3, 19/5, 29/7, 43/11, 53/13, 67/17. For example, if p and q are primes and q > 13, then 53/13 is closer to 3 than p/q is.

Cf. A000040, A023212, A062737, A090866, A120639, A162857.

x = 4; 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, A265765/A120639 *)
Numerator[tL]   (* A162857 *)
Denominator[tL] (* A062737 *)
Numerator[tU]   (* A090866 *)
Denominator[tU] (* A023212 *)
Numerator[y]    (* A265765 *)
Denominator[y]  (* A120639 *)

A164569 Primes p such that 11*p+8 are prime numbers.

Original entry on oeis.org

3, 13, 31, 73, 79, 151, 163, 181, 193, 241, 283, 349, 373, 379, 409, 421, 463, 601, 631, 673, 751, 769, 811, 829, 853, 883, 991, 1021, 1039, 1063, 1171, 1201, 1303, 1381, 1423, 1429, 1453, 1459, 1471, 1543, 1549, 1579, 1609, 1621, 1663, 1669, 1789, 1801

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Apart from the first term, a(n) = 1 (mod 6).

			11*3+8=41, ..

Cf. A023212, A023217, A023224, A023230, A023239

lst={};Do[p=Prime[n];If[PrimeQ[11*p+8],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[500]],PrimeQ[11#+8]&] (* Harvey P. Dale, Jul 17 2011 *)

Comment from Charles R Greathouse IV, Oct 12 2009

A164570 Primes p such that 8p-3 and 8p+3 are also prime numbers.

Original entry on oeis.org

2, 5, 7, 13, 47, 103, 107, 127, 163, 233, 293, 337, 383, 433, 443, 467, 503, 673, 677, 733, 797, 877, 1087, 1093, 1153, 1217, 1223, 1307, 1637, 1933, 2053, 2087, 2137, 2423, 2477, 2543, 2633, 2687, 2857, 2917, 3163, 3373, 3407, 3467, 3767, 3793, 3877

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Subsequence of A023229. [R. J. Mathar, Aug 26 2009]

Primes of the form A087695(k)/8. [R. J. Mathar, Aug 26 2009]

			For p=2, 8*2-3=13 and 8*2+3=19 are prime numbers, which adds p=2 to the sequence
For p=5, 8*5-3=37 and 8*5+3=43 are prime numbers, which adds p=5 to the sequence.

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

Cf. A023212, A023217, A023224, A023230, A023239, A060212, A106015, A124098, A125272, A127430, A153768, A164566, A164567, A164568, A164569.

[p: p in PrimesUpTo(3000) | IsPrime(8*p-3) and IsPrime(8*p+3)]; // Vincenzo Librandi, Apr 09 2013

lst={};Do[p=Prime[n];If[PrimeQ[8*p-3]&&PrimeQ[8*p+3],AppendTo[lst,p]], {n,7!}];lst
Select[Prime[Range[1000]], And@@PrimeQ/@{8 # + 3, 8 # - 3}&] (* Vincenzo Librandi, Apr 09 2013 *)
Select[Prime[Range[1000]],AllTrue[8#+{3,-3},PrimeQ]&] (* Harvey P. Dale, May 05 2023 *)

Comments turned into examples by R. J. Mathar, Aug 26 2009

A186721 Numbers n such that n, 4n + 1 and 6n + 1 are all prime.

Original entry on oeis.org

3, 7, 13, 37, 73, 277, 373, 577, 727, 853, 1033, 1063, 1327, 1423, 1453, 1567, 1777, 1987, 2293, 2677, 2767, 3037, 3163, 3307, 3457, 4273, 4447, 4993, 5197, 5557, 6247, 6673, 7573, 8353, 8893, 9067, 9397, 9463, 9547, 9613, 10303, 10903, 12007, 12973, 13177, 14083

Offset: 1

Zak Seidov, Jan 21 2012

nonn

Subsequence of A023212.

Cf. A023212.

Join[{3},Select[Range[7, 20000, 6], PrimeQ[#] && PrimeQ[4# + 1] && PrimeQ[6# + 1] &]]

A228857 Odd primes p > 3 for which 14*p+1 is also prime.

Original entry on oeis.org

5, 17, 47, 53, 59, 83, 107, 113, 149, 167, 173, 239, 269, 353, 419, 443, 449, 503, 509, 563, 587, 599, 647, 659, 677, 719, 797, 827, 929, 947, 977, 983, 1097, 1103, 1109, 1187, 1193, 1223, 1229, 1259, 1289, 1367, 1409, 1427, 1433, 1439, 1493, 1523, 1667

Offset: 1

Ant King, Sep 06 2013

nonn

In 1823, Legendre proved that the first case of Fermat’s Last Theorem is true for all exponents that are members of this sequence (see Ribenboim’s reference, p.112).

			As both 5 and 14*5 + 1 = 71 are prime, then 5 is a member of this sequence.

Paulo Ribenboim; Fermat’s Last Theorem For Amateurs, Springer-Verlag, New York, Inc., (1999).

Cf. A005384, A023212, A023228, A023237, A155943.

[p: p in PrimesInInterval(5,2000) |IsPrime(14*p+1)]; // Vincenzo Librandi, Sep 18 2016

Select[Prime[Range[3,1667]],PrimeQ[14#+1] &]

lista(nn) = forprime(p=5, nn, if(isprime(14*p+1), print1(p, ", "))); \\ Altug Alkan, Sep 18 2016

A297306 Primes p such that q = 4p+1 and r = (2p+1)/3 are also primes.

Original entry on oeis.org

7, 43, 79, 163, 673, 853, 919, 1063, 1429, 1549, 1663, 2143, 2683, 3229, 3499, 4993, 5119, 5653, 5779, 6229, 6343, 7333, 7459, 7669, 8353, 8539, 8719, 9829, 10009, 10243, 10303, 11383, 11689, 12583, 13399, 14149, 14653, 14923, 15649, 16603, 17053, 17389, 17749

Offset: 1

David S. Newman, Jan 04 2018

nonn

This sequence was suggested by Moshe Shmuel Newman. It has its source in his study of finite groups.

			Prime p = 7 is in the sequence because q = 4*7+1 = 29 and r = (2*7+1)/3 = 5 are also primes.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040.

Intersection of A023212 and A104163.

a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p); if irem(p, 3)=1 and
         isprime(4*p+1) and isprime((2*p+1)/3) then break fi
      od; p
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 07 2018

a[n_] := a[n] = Module[{p}, p = If[n == 1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[Mod[p, 3] == 1 && PrimeQ[4p+1] && PrimeQ[(2p+1)/3], Break[]]]; p];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

isok(p) = isprime(p) && isprime(4*p+1) && iferr(isprime((2*p+1)/3), E, 0); \\ Michel Marcus, Nov 27 2020

More terms from Alois P. Heinz, Jan 07 2018

A318251 Lesser of amicable numbers pair (m, n) such that n = H(m) and m = H(n) where H(n) = A074206(n) is the number of ordered factorizations of n.

Original entry on oeis.org

6144, 19329024, 939524096, 4026531840, 309237645312, 6146186280960, 52158082842624, 29273397577908224

Offset: 1

Amiram Eldar, Aug 22 2018

The larger numbers in each pair are in A318252.
Analogous to A002025 as A163272 is analogous to A000396.
If p and 4p+1 are primes then 2^(4p-1)*p is in this sequence, therefore if A023212 is infinite then also this sequence is.
The terms were calculated using an extended list of terms of A025487.

			6144 is in the sequence since A074206(6144) = 13312 and A074206(13312) = 6144.

Cf. A023212, A074206, A090866, A163272, A318252.

f(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, f(A[k])));
isok(n)={my(a=f(n)); a>n && f(a)==n;} \\ Michel Marcus, Sep 26 2018

cp's OEIS Frontend

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

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A087634 Primes p such that the equation phi(x) = 4p has a solution, where phi is the totient function.

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A182434 Number of primes p < n such that 4*p+1 is also prime.

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A265765 Numerators of primes-only best approximates (POBAs) to 4; see Comments.

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Mathematica

A164569 Primes p such that 11*p+8 are prime numbers.

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Mathematica

Extensions

A164570 Primes p such that 8*p-3 and 8*p+3 are also prime numbers.

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Magma

Mathematica

Extensions

A186721 Numbers n such that n, 4n + 1 and 6n + 1 are all prime.

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Mathematica

A228857 Odd primes p > 3 for which 14*p+1 is also prime.

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Magma

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PARI

A164570 Primes p such that 8p-3 and 8p+3 are also prime numbers.

A297306 Primes p such that q = 4p+1 and r = (2p+1)/3 are also primes.