

A158460


Numbers n such that prime(n1)+5 is square and equal to prime(n+1)1.


4



6, 44, 41202, 123125, 141582, 527569, 920270, 975866, 1034000, 1278000, 1504258, 1707305, 1774017, 1863515, 2513332, 2776350, 3315370, 5100781, 5152209, 5746269, 5943102, 7380924, 7891751, 8585974, 10100295, 11022570, 12248841, 13213333, 13654151, 13817964
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OFFSET

1,1


COMMENTS

Conjecture: If the condition holds, prime(n1) and prime(n) are twin primes of the form 10k+1 and 10k+3. Ie., the last digits are 1 and 3.
This is so because prime(n1)+5 square => possible ending digits of 0,1,4,5,6,9. To get ending digits for prime(n1) we subtract 5 to get ending digits 5,6,9,0,1,4. So 1,9 are the only possible endings since 0,4,5,6 => prime(n1) not prime, impossible. Now by the condition of equality, prime(n1)+6 = prime(n+1). So if prime(n1) ends in 9, prime(n1) +6 ends in 5 => prime(n+1) not prime, impossible. Therefore prime(n1) ends in 1 and by the condition of prime(n1) and prime(n) being twin primes, prime(n) ends in 3. This sequence is a variation of the conjecture provided in the link. The Pari script allows for general investigation of numbers of the form prime(n1)+a and prime(n+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(n1) retrieval before calling the second prime(n+1). This cuts the search time in half. A much faster and more inclusive program is in the Link. This gcc program and the 300 gig file computes a(n) up to n = 37 billion.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Cino Hilliard, Integer equal Gcc Program
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.


FORMULA

Prime(n) is the nth prime number.


EXAMPLE

For n=6, prime(61) = 11, 11+5 = 16 = prime(6+1)1 = 171 so 6 is the first entry in the sequence.


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

Cf. A158509.
Sequence in context: A267074 A271963 A065783 * A077672 A119202 A286325
Adjacent sequences: A158457 A158458 A158459 * A158461 A158462 A158463


KEYWORD

nonn


AUTHOR

Cino Hilliard, Mar 19 2009


EXTENSIONS

More terms from Alois P. Heinz, Sep 07 2016


STATUS

approved



