A158529 List of primes p with following properties: p = prime(n-1) for some n, p+7 is a square and is equal to prime(n+1)-1.

29, 569, 1289, 41609, 147449, 2322569, 2842589, 7096889, 7485689, 10074269, 16208669, 21288989, 33802589, 54819209, 56610569, 57699209, 59814749, 115218749, 118069949, 126427529, 134235389, 149670749, 196448249, 240746249

Offset

1,1

Comments

Conjecture: If the condition holds, prime(n-1) and prime(n) are twin primes of the form 10k+9 and 10k1+1, i.e. the last digits of the twin prime pairs are 9 and 1. The 9 ending is evident in this sequence. The table of the first 101 terms was computed using Zak Seidov's table.

Links

Cino Hilliard, List of n, a(n) for n=1..101

S. M. Ruiz, Integer then equal.

Sebastian Martin Ruiz and others, Integers then Equals, digest of 7 messages in primenumbers Yahoo group, Mar 14 - Mar 20, 2009.

Zak Seidov, A158470 first 101 terms.

Formula

Prime(n) is the n-th prime number.

Example

For n = 11, prime(11-1)=29, 29+7=36; prime(11+1)=37, 37-1=36. So 29 is the first entry in the sequence.

Mathematica

ppQ[{a_, b_}]:=Module[{s=Prime[a+1]-1}, IntegerQ[Sqrt[s]]&&b+7==s]; Select[ Table[ {n, Prime[n-1]}, {n, 2, 133*10^5}], ppQ][[All, 2]] (* Harvey P. Dale, Jul 31 2020 *)

Prog

(PARI) \\Copy and paste the Zak's file to zaklist.txt and edit to a straight
\\list with CR after each entry. Start a new Pari sesion then \r zakilist.txt
integerequal(a, b) =
{
local(x, p1, p2);
for(j=1, 101,
x=eval(concat("%", j)); p1=prime2(x-1);
if(issquare(p1+a),
p2=prime2(x+1); if((p1+a)==(p2-b),
print1(p1", ")
)
prime2(n) = \\the n-th prime using c:\sieve\prime.exe calling 8byte binary
\\g:\sievedata\prime2-1trill.bin" 300 gig file of primes <10^12
{
local(x, s);
s=concat("c:/sieve/prime ", Str(n));
s=concat(s, " > temp.txt");
\\Must save to a temp file for correct output
system(s);
return(read("temp.txt"))
}
)
)
}

Crossrefs

Sequence in context: A023948 A020974 A167740 * A020766 A069295 A103723

Adjacent sequences: A158526 A158527 A158528 * A158530 A158531 A158532

Keyword

nonn

Author

Cino Hilliard, Mar 20 2009

Extensions

Edited by N. J. A. Sloane Aug 31 2009 (rephrased definition, corrected offset).

Status

approved