A059324 -id:A059324 - cp's OEIS frontend

A138479 a(n) = smallest prime p such that 2*n + p^2 is another prime, or 0 if no such prime exists.

3, 3, 5, 3, 3, 5, 3, 5, 5, 3, 3, 7, 0, 3, 7, 3, 3, 5, 3, 7, 5, 3, 5, 5, 3, 3, 5, 0, 3, 7, 3, 3, 29, 0, 3, 5, 3, 5, 5, 3, 5, 5, 0, 3, 7, 3, 3, 19, 3, 3, 5, 3, 5, 7, 0, 5, 5, 0, 3, 11, 3, 5, 5, 3, 3, 5, 0, 11, 5, 3, 3, 7, 0, 3, 7, 0, 3, 5, 3, 11, 7, 3, 5, 5, 3, 3, 5, 0, 7, 7, 3, 3, 5, 3, 3, 7, 0, 11, 5, 0

Offset: 1

Philippe LALLOUET (philip.lallouet(AT)orange.fr), Mar 20 2008

nonn

For numbers k such that a(k) = 0 see A138685.

			11=2+3^2 hence a(1)=3,
13=4+3^2 hence a(2)=3,
31=6+5^2 hence a(3)=5.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Near-Square Prime

Cf. A002373, A020481, A049613, A059324 (?).

a:= proc(n) local p;
      if irem(n, 3)=1 and not isprime(2*n+9) then 0
    else p:=2;
         do p:= nextprime(p);
            if isprime(2*n+p^2) then return p fi
         od
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 16 2014

a = {}; Do[ p = 0; While[ (! PrimeQ[ 2*n + Prime[ p + 1 ]2 ]) && (p < 1000), p++ ]; If[ p < 1000, AppendTo[ a, Prime[ p + 1 ] ], AppendTo[ a, 0 ] ], {n, 1, 150} ]; a (* Artur Jasinski, Mar 26 2008 *)
a[n_]:=If[Mod[n,3]!=1,(For[m=1,!PrimeQ[2n+Prime[m]^2],m++ ]; Prime[m]),If[ !PrimeQ[2n+9],0,3]];Table[a[n],{n,100}] (* Farideh Firoozbakht, Mar 28 2008 *)

More terms from Artur Jasinski and Farideh Firoozbakht, Mar 26 2008

A059325 Numbers n such that 6n + 5 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 21, 22, 24, 27, 28, 29, 31, 32, 37, 38, 39, 41, 42, 43, 44, 46, 48, 51, 52, 57, 58, 59, 63, 64, 66, 69, 71, 73, 74, 76, 77, 79, 81, 83, 84, 86, 92, 93, 94, 97, 98, 99, 102, 106, 107, 108, 109, 112, 113, 116, 119, 123, 126

Offset: 1

Anton Joha, Jan 26 2001

			a(4)=3 because 6*3 + 1 = 19 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Complement of A059324.

[n: n in [0..200]| IsPrime(6*n+5)] // Vincenzo Librandi, Aug 07 2010

Select[Range[0, 1000], PrimeQ[6 # + 5] &] (* G. C. Greubel, Jan 06 2017 *)

is(n)=isprime(6*n+5) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Henry Bottomley, Jan 29 2001

A153170 Numbers k such that 3*k + 2 is not prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 38, 39, 40, 41, 42, 44, 46, 47, 48, 50, 51, 52, 53, 54, 56, 58, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 91, 92, 94, 95, 96, 98, 99, 100, 101, 102

Offset: 1

Vincenzo Librandi, Dec 20 2008

Contains the positive even numbers (A005843) and the odd numbers of the form 2*A059324(.) + 1. - R. J. Mathar, Nov 27 2010

Numbers k such that (3*k)!/(3*k + 2) is an integer. - Peter Bala, Jan 25 2017

			Distribution of the odd terms in the following triangular array:
  *;
  *,   *;
  *,  11,   *;
  *,   *,   *,   *;
  *,   *,  25,   *,   *;
  *,  21,   *,   *,  47,   *;
  *,   *,   *,   *,   *,   *,   *;
  *,   *,  39,   *,   *,  73,   *,   *;
  *,  31,   *,   *,  69,   *,   *, 107,   *;
  *,   *,   *,   *,   *,   *,   *,   *,   *,   *;
  *,   *,  53,   *,   *,  99,   *,   *, 145,   *,   *;
  *,  41,   *,   *,  91,   *,   *, 141,   *,   *, 191,   *;
etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h - 1)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 15 2013

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

Cf. A024893, A014076, A045751, A095277, A153088, A153329, A153343.

[n: n in [1..110] | not IsPrime(3*n + 2)]; // Vincenzo Librandi, Oct 11 2012

for n from 0 to 100 do
if irem(factorial(3*n), 3*n+2) = 0 then print(n); end if;
end do: # Peter Bala, Jan 25 2017

Select[Range[1, 200], !PrimeQ[3*# + 2] &] (* Vincenzo Librandi, Oct 11 2012 *)

for(n=1,200,if(!isprime(3*n+2), print1(n,", "))) \\  Joerg Arndt, Nov 27 2010

Edited by N. J. A. Sloane, Jun 23 2010

A094517 Primes p such that 6p+11 is not a prime.

Original entry on oeis.org

11, 19, 29, 53, 59, 61, 67, 71, 79, 89, 103, 109, 113, 137, 139, 149, 151, 163, 179, 191, 193, 199, 223, 229, 239, 241, 251, 263, 269, 277, 293, 307, 337, 347, 349, 359, 373, 379, 389, 409, 419, 421, 431, 439, 443, 449, 467, 479, 487, 499, 503, 509, 523, 547, 557, 569

Offset: 1

N. J. A. Sloane, following a suggestion by R. K. Guy, Jun 05 2004

nonn

Primes of the form A059324(.)-1. - R. J. Mathar, Jan 30 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Complement in primes of A089438.

[p: p in PrimesUpTo(5000)|not IsPrime(6*p+11)]; // Vincenzo Librandi, Jan 30 2011

Select[Prime[Range[150]],CompositeQ[6#+11]&] (* Harvey P. Dale, Aug 02 2021 *)

A275237 Smallest number k > 0 such that sigma(x) and sigma(x)+2 are both prime, where x = (6k+1)^(6n+4), or -1 if no such k exists.

Original entry on oeis.org

1, 348, 436, 6018, 5880, -1, 4612, 26921, 16166, 81111, -1, 426260, -1, 181876, 227180, -1, 12836, 287388, 2317, -1, -1, 1128403, 668927, -1, 5295, -1, -1, 490118, 2217967, 1607226, -1, 1212183, 100728, -1, -1, -1, -1, 1191713, 43475567, 165965, -1, 2915491, 361885, 4159496, 3398061, -1, 88930, -1, 10451327, -1, -1

Offset: 0

Chai Wah Wu, Jul 20 2016

If x is a number such that sigma(x) and sigma(x)+2 are both prime (A274962), then x = 2 or x is of the form (6k+1)^(6r+4) where 6k+1 is prime.
For p = 6*k+1, sigma(p^34) = (46656*k^6 + 54432*k^5 + 27216*k^4 + 7560*k^3 + 1260*k^2 + 126*k + 7) * (1296*k^4 + 1080*k^3 + 360*k^2 + 60*k + 5) * c(k), thus a(5) = -1. - Altug Alkan , Jul 21 2016
Similarly a(12) = a(19) = a(23) = a(26) = a(33) = a(34) = -1. Furthermore, for all r > 0, a(5*r) = -1 since sigma((6k+1)^(30r+4)) = ((6*k+1)^(6*r) + ((6*k+1)^(6*r) -1)/(6*k))*(1296*k^4*(6*k + 1)^(24*r) + 864*k^3*(6*k + 1)^(24*r) + 216*k^3*(6*k + 1)^(18*r) + 216*k^2*(6*k + 1)^(24*r) + 108*k^2*(6*k + 1)^(18*r) + 36*k^2*(6*k + 1)^(12*r) + 24*k*(6*k + 1)^(24*r) + 18*k*(6*k + 1)^(18*r) + 12*k*(6*k + 1)^(12*r) + 6*k*(6*k + 1)^(6*r) + (6*k + 1)^(24*r) + (6*k + 1)^(18*r) + (6*k + 1)^(12*r) + (6*k + 1)^(6*r) + 1). - Chai Wah Wu, Jul 21 2016

			For n = 0, x = 7^4 is the smallest fourth power such that sigma(x) and sigma(x)+2 are both prime, thus a(0) = 1.

Cf. A000203, A023194, A249763, A274962, A274963.

a(A059324(n)) = -1. - Altug Alkan, Aug 13 2016

a(31)-a(37) from Chai Wah Wu, Aug 01 2016

a(38)-a(50) from Chai Wah Wu, Aug 18 2016

cp's OEIS Frontend

A138479 a(n) = smallest prime p such that 2*n + p^2 is another prime, or 0 if no such prime exists.

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A059325 Numbers n such that 6n + 5 is prime.

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Magma

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A153170 Numbers k such that 3*k + 2 is not prime.

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Magma

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A094517 Primes p such that 6p+11 is not a prime.

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Magma

Mathematica

A275237 Smallest number k > 0 such that sigma(x) and sigma(x)+2 are both prime, where x = (6k+1)^(6n+4), or -1 if no such k exists.

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