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A158460 Numbers n such that prime(n-1)+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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: If the condition holds, prime(n-1) 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(n-1)+5 square => possible ending digits of 0,1,4,5,6,9. To get ending digits for prime(n-1) 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(n-1) not prime, impossible. Now by the condition of equality, prime(n-1)+6 = prime(n+1). So if prime(n-1) ends in 9, prime(n-1) +6 ends in 5 => prime(n+1) not prime, impossible. Therefore prime(n-1) ends in 1 and by the condition of prime(n-1) 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(n-1)+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(n-1) 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 n-th prime number.

EXAMPLE

For n=6, prime(6-1) = 11, 11+5 = 16 = prime(6+1)-1 = 17-1 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(x-1);

if(issquare(p1+a),

p2=prime(x+1); if((p1+a)==(p2-b),

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

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Last modified December 3 14:22 EST 2020. Contains 338906 sequences. (Running on oeis4.)