A165352 -id:A165352 - cp's OEIS frontend

A165353 Primes p such that p+(p^2-1)/8 is a prime number.

7, 17, 23, 31, 41, 71, 73, 79, 103, 113, 127, 151, 191, 233, 257, 281, 311, 337, 367, 383, 409, 433, 463, 487, 569, 577, 593, 601, 607, 647, 809, 887, 911, 953, 977, 1039, 1049, 1087, 1097, 1153, 1193, 1201, 1223, 1279, 1297, 1321, 1361, 1399, 1433, 1481, 1487

Offset: 1

Vincenzo Librandi, Sep 16 2009

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

Cf. A165352, A165354.

[n: n in [3..1500] |IsPrime(n) and IsPrime(n+(n^2-1) div 8)]; // Vincenzo Librandi, Sep 01 2012

Select[Prime[Range[250]], PrimeQ[# + (#^2 - 1)/8] &] (* Vincenzo Librandi, Sep 01 2012 *)

a(n)+(a(n)^2-1)/8 = A165352(n).

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

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A165353 Primes p such that p+(p^2-1)/8 is a prime number.

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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

Mathematica

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

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Mathematica

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