A165353 -id:A165353 - cp's OEIS frontend

A165352 Primes of the form p + (p^2 - 1)/8, where p is also prime.

13, 53, 89, 151, 251, 701, 739, 859, 1429, 1709, 2143, 3001, 4751, 7019, 8513, 10151, 12401, 14533, 17203, 18719, 21319, 23869, 27259, 30133, 41039, 42193, 44549, 45751, 46663, 52973, 82619, 99233, 104651, 114479, 120293, 135979, 138599, 148783, 151523

Offset: 1

Vincenzo Librandi, Sep 16 2009

Primes of the form k+(k^2-1)/8, where k >0 are prime or composite, are the superset 13, 19, 43, 53, 89, 103 etc.

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

Cf. A165353, A165354.

[a: p in PrimesInInterval(3, 1200) | IsPrime(a) where a is p + (p^2 - 1) div 8 ]; // Vincenzo Librandi, Oct 12 2012

Select[Table[p + (p^2 - 1)/8, {p, Prime[Range[200]]}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

forprime(p=3,1e4,if(isprime(t=p^2>>3+p),print1(t", "))) \\ Charles R Greathouse IV, May 15 2013

a(n) = A165353(n)+(A165353(n)^2-1)/8.

More terms from R. J. Mathar, Sep 21 2009

A165354 Primes p such that p+(p^2-1)/8 is an integer but not a prime number.

Original entry on oeis.org

3, 5, 11, 13, 19, 29, 37, 43, 47, 53, 59, 61, 67, 83, 89, 97, 101, 107, 109, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 263, 269, 271, 277, 283, 293, 307, 313, 317, 331, 347, 349, 353, 359, 373, 379, 389

Offset: 1

Vincenzo Librandi, Sep 16 2009

For p=2, p+(p^2-1)/8 is not integer; for all others, p=4k+-1, it is an integer.

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

Cf. A165352, A165353.

[p: p in PrimesUpTo(500)| not IsPrime(Floor(p+(p^2-1)/8))]; // Vincenzo Librandi, Sep 12 2013

Select[Prime[Range[2, 500]], ! PrimeQ[(#^2 - 1) / 8 + #]&] (* Vincenzo Librandi, Sep 12 2013 *)

A000040 \ {{2} U A165353}. - R. J. Mathar, Sep 21 2009

Entries checked by R. J. Mathar, Sep 21 2009

A165356 Primes p such that p + (p^2 - 1)/8 is a perfect square.

Original entry on oeis.org

3, 19, 211, 1249, 4513, 1445953, 30381331, 286292179, 2959257735801707821729

Offset: 1

Vincenzo Librandi, Sep 16 2009

The primes p = A000040(j) at j= 2, 8, 47, 204, 612, 110340 etc. generating the squares 2^2, 8^2, 76^2, 443^2 etc.

From the ansatz p + (p^2 - 1)/8 = s^2 we conclude p = -4 + sqrt(17 + 8*s^2), so all s are members of A077241.

			For p=3, p + (p^2-1)/8 = 4 = 2^2. For p=19, p + (p^2-1)/8 = 64 = 8^2. For p=211, p + (p^2-1)/8 = 5776 = 76^2.

Cf. A165352, A165353, A165354.

A077241 := proc(n) if n <= 3 then op(n+1,[1,2,8,13]) ; else 6*procname(n-2)-procname(n-4) ; fi; end:
for n from 0 do s := A077241(n) ; p := sqrt(17+8*s^2)-4 ; if isprime(p) then printf("%d,\n",p) ; fi; od: # R. J. Mathar, Sep 21 2009
a := proc (n) if isprime(n) = true and type(sqrt(n+(1/8)*n^2-1/8), integer) = true then n else end if end proc; seq(a(n), n = 1 .. 10000000); # Emeric Deutsch, Sep 21 2009

p = 2; lst = {}; While[p < 10^12, If[ IntegerQ@ Sqrt[p + (p^2 - 1)/8], AppendTo[lst, p]; Print@p]; p = NextPrime@p] (* Robert G. Wilson v, Sep 30 2009 *)

6 more terms from R. J. Mathar, Sep 21 2009

cp's OEIS Frontend

A165352 Primes of the form p + (p^2 - 1)/8, where p is also prime.

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A165354 Primes p such that p+(p^2-1)/8 is an integer but not a prime number.

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Magma

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A165356 Primes p such that p + (p^2 - 1)/8 is a perfect square.

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