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A138511
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Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q.
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18
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10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 291, 295, 298, 302, 303, 305
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OFFSET
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1,1
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COMMENTS
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From Antti Karttunen, Dec 17 2014, further edited Jan 01 & 04 2014: (Start)
Semiprimes p*q, p < q, such that the smallest r for which r^k <= p and q < r^(k+1) [for some k >= 0] is q+1, and thus k = 0. In other words, semiprimes whose both prime factors do not fit (simultaneously) between any two consecutive powers of any natural number r less than or equal to the larger prime factor. This condition forces the larger prime factor q to be greater than the square of the smaller prime factor because otherwise the opposite condition given in A251728 would hold.
Assuming that A054272(n), the number of primes in interval [prime(n), prime(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true), these are also "unsettled" semiprimes that occur in a square array A083221 constructed from the sieve of Eratosthenes, "above the line A251719", meaning that if and only if row < A251719(col) then a semiprime occurring at A083221(row, col) is in this sequence, and conversely, all the semiprimes that occur at any position A083221(row, col) where row >= A251719(col) are in the complementary sequence A251728.
(End)
Semiprimes p*q, p < q, such that b = q+1 is the minimal base with the property that p and q have equal length representations in base b. This was the original definition, which is based primarily on A138510: A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.
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LINKS
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FORMULA
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Other identities. For all n >= 1 it holds that:
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EXAMPLE
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PROG
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(Haskell)
a138511 n = a138511_list !! (n-1)
a138511_list = filter f [1..] where
f x = p ^ 2 < q && a010051' q == 1
where q = div x p; p = a020639 x
(PARI) isok(s) = my(f=factor(s)); (bigomega(f) == 2) && (#f~ == 2) && (f[1, 1]^2 < f[2, 1]); \\ Michel Marcus, Sep 15 2020
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CROSSREFS
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Also an intersection of A001358 and A253569, from the latter which this sequence differs for the first time at n=60, where A253569(60) = 290, while here a(60) = 291.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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New definition by Antti Karttunen, Jan 01 2015; old definition moved to comment.
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STATUS
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approved
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