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 A082130 Numbers k such that 2*k-1 and 2*k+1 are semiprimes. 4
 17, 25, 28, 43, 46, 47, 60, 61, 71, 72, 80, 92, 93, 101, 102, 107, 108, 109, 110, 118, 124, 133, 144, 145, 150, 151, 152, 160, 161, 164, 170, 196, 197, 206, 207, 208, 223, 226, 235, 236, 258, 259, 264, 267, 268, 272, 276, 290, 291, 295, 317, 334, 335, 340, 343, 344 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Let p and q be distinct odd primes, and take a and b so that a*p - b*q = 1. Dickson's conjecture implies there are infinitely many k such that 2*a+k*q and 2*b+k*p are prime, in which case n = a*p + (k*q*p-1)/2 is in the sequence with 2*n-1 = (2*b+k*p)*q and 2*n+1 = (2*a+k*q)*p. - Robert Israel, Aug 13 2018 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Wikipedia, Dickson's conjecture. EXAMPLE 17 is a term because 2*17 - 1 = 33 = 3*11 and 2*17 + 1 = 35 = 5*7 are both semiprimes. MAPLE OSP:= select(numtheory:-bigomega=2, {seq(i, i=3..1000, 2)}): R:= map(t -> (t+1)/2, OSP intersect map(`-`, OSP, 2)): sort(convert(R, list)); # Robert Israel, Aug 13 2018 PROG (PARI) isok(n) = (bigomega(2*n-1) == 2) && (bigomega(2*n+1) == 2); \\ Michel Marcus, Jul 16 2017 (Python) from sympy import factorint from itertools import count, islice def agen(): # generator of terms nxt = 0 for k in count(2, 2): prv, nxt = nxt, sum(factorint(k+1).values()) if prv == nxt == 2: yield k//2 print(list(islice(agen(), 56))) # Michael S. Branicky, Nov 26 2022 CROSSREFS Cf. A001358, A082131. Sequence in context: A272635 A105448 A336007 * A140609 A131275 A227238 Adjacent sequences: A082127 A082128 A082129 * A082131 A082132 A082133 KEYWORD nonn AUTHOR Hugo Pfoertner, Apr 04 2003 EXTENSIONS More terms from Jud McCranie, Apr 04 2003 STATUS approved

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Last modified June 18 13:40 EDT 2024. Contains 373481 sequences. (Running on oeis4.)