

A158470


Numbers k such that prime(k1) + 7 is square and equal to prime(k+1)  1.


3



11, 105, 210, 4352, 13631, 171030, 206287, 482817, 507376, 669211, 1043655, 1347091, 2078002, 3272095, 3372558, 3433588, 3551781, 6584471, 6738010, 7186808, 7604542, 8426927, 10893207, 13200411, 15175773, 23350193, 25653343
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OFFSET

2,1


COMMENTS

If the condition holds, prime(k1) and prime(k) are twin primes. These are of the form 10m+9 and 10m+1. I.e., the last digits are 9 and 1. This is true because a square number must end in 0,1,4,5,6,9. So prime(k1)+7 is square => it must end in one of these numbers. So to find the ending of prime(k1), we subtract 7 from 0,1,4,5,6,9 to get the last digit, i.e., 3,4,7,8,9,2.
Since prime(k1) is prime, endings 2,4,5,8 are not allowed. This leaves us with 3,7,9 as possible endings of prime(k1). Now to get prime(k), which the condition states is 2 greater than prime(k1), we add 2 and 3+2=5 => prime(k) not prime, impossible. So the possible endings of prime(k1) are reduced to 7 or 9. Now the condition prime(k1)+7 = prime(k+1)1 => prime(k1)+8 = prime(k+1). Then adding 7 => prime(k+1) ends in 5, impossible. So prime(k1) must end in 9, and adding 2 makes prime(k) end in 1. This sequence is a calculation of the conjecture provided in the link. The PARI script provides for the general investigation of numbers of the form prime(k1)+a and prime(k+1)b. The values a=5,7; b=1 consistently yield twin primes when the condition holds.
Notice we test for square of the first prime(k1) retrieval before calling the second prime(k+1). This cuts the search time in half. A far superior search routine is the C program found in the link, which reads a huge 300 GB file of the primes < 1 trillion + 1 billion.


LINKS

Table of n, a(n) for n=2..28.
Cino Hilliard, C program (broken link) [From Cino Hilliard, Mar 21 2009]
S. M. Ruiz, Integer 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 100 terms


EXAMPLE

For k=11, prime(111) = 29, 29+7 = 36 = prime(11+1)1 = 371 so 11 is the first entry in the sequence.


MATHEMATICA

pnsQ[n_]:=Module[{p1=Prime[n1], p2=Prime[n+1]}, p1+7==p21&&IntegerQ[ Sqrt[ p1+7]]]; Select[Range[5, 25660000], pnsQ]


PROG

(PARI) integerequal(m, n, a, b) =
{
local(x, p1, p2);
for(x=m, n,
p1=prime(x1);
if(issquare(p1+a),
p2=prime(x+1); if((p1+a)==(p2b),
print(x", "p1", "prime(x))
)
)
)
}


CROSSREFS

Sequence in context: A173851 A358340 A295840 * A163933 A359987 A099839
Adjacent sequences: A158467 A158468 A158469 * A158471 A158472 A158473


KEYWORD

nonn


AUTHOR

Cino Hilliard, Mar 19 2009


EXTENSIONS

More terms from Zak Seidov, Mar 20 2009


STATUS

approved



