A336585 Integers m such that (m-d)*(m+d) < (m-1)^2, where d is the smallest number such that both m-d and m+d are primes.

22, 28, 32, 38, 46, 49, 55, 58, 68, 74, 82, 87, 94, 112, 121, 128, 130, 136, 146, 155, 184, 200, 203, 206, 218, 221, 224, 238, 244, 247, 253, 265, 268, 284, 286, 301, 304, 306, 308, 316, 318, 320, 323, 326, 341, 344, 346, 362, 398, 412, 413, 428, 454, 466, 484

Offset

1,1

Comments

All terms in this sequence are composites since, if m is a prime, d = 0 and (m-d)*(m+d) = m^2 > (m-1)^2.

It seems that the number of terms in this sequence is finite, with the last term being a(1225) = 1353559. Conjecture: there exist only 1225 semiprimes of the form (m-d)*(m+d), where d is the smallest number such that (m-d)*(m+d) < (m-1)^2.

a(n) in this sequence is the value of n in A047160 with m > sqrt(2*n - 1).

All terms <= 1353559 in A335297 can be found in this sequence.

Links

Table of n, a(n) for n=1..55.

Example

2 is not a term since for m = 2, d = 0 and (2-0)*(2-0) = 4 > (m-1)^2 = 1;
4 is not a term since for m = 4, d = 1 and (4-1)*(4+1) = 15 > (m-1)^2 = 9;
22 is a term since for m = 22, d = 9 and (22-9)*(22+9) = 403 < (m-1)^2 = 441;
1353559 is a term since for m = 1353559, d = 1722 and (1353559-1722)*(1353559+1722) = 1832119001197 < (m-1)^2 = 1832119259364.

Prog

(Python)
from sympy import isprime
n = 0
m = 2
while m >= 2:
    d = 0
    while d < m/2:
        p = m - d
        q = m + d
        if isprime(p) == 1 and isprime(q) == 1:
            if p*q < (m - 1) * (m - 1):
                n += 1
                print (m)
            break
        d += 1
    m += 1

Crossrefs

Cf. A047160, A335297.

Sequence in context: A043908 A177679 A276184 * A255222 A225337 A298588

Adjacent sequences: A336582 A336583 A336584 * A336586 A336587 A336588

Keyword

nonn

Author

Ya-Ping Lu, Oct 04 2020

Status

approved