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A336585
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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