cp's OEIS Frontend

A084142 Positive numbers k such that the number of primes between k and 2*k is different from the number of primes between m and 2*m for every number m != k.

%I A084142 #16 Feb 16 2025 08:32:49
%S A084142 1,120,216,300,531,714,804,999,1344,1356,1395,1680,1764,1770,1959,
%T A084142 2046,2121,2325,2484,2511,2760,2826,3150,3285,3396,3744,4044,4116,
%U A084142 4146,4314,4710,4839,5046,5070,5136,5250,5586,5970,6411,6459,6501,6504,6846,7275
%N A084142 Positive numbers k such that the number of primes between k and 2*k is different from the number of primes between m and 2*m for every number m != k.
%C A084142 The number of primes between k and 2*k is unique because no other number m > 0 has the same of primes between m and 2m, exclusively. k is the value of A060756(j) or A084139(j) when A084138(j) = 1. Question: Is this sequence infinitely long?
%C A084142 If k > 1 is a term then A060715(k-1) < A060715(k) < A060715(k+1). Consequently, (2*k-1, 2*k+1) is a twin prime pair, so 3|k. Additionally, it can be shown that k-1..k+3 are all composite numbers. Moreover, if k is even, then k-4..k+4 are all composite numbers. - _Jon E. Schoenfield_, Oct 08 2023
%D A084142 P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.
%H A084142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate</a>.
%e A084142 120 is a term because there are 22 primes between 120 and 240 and no other number m > 0 has 22 primes between m and 2*m.
%Y A084142 Cf. A060715, A060756, A084138, A084139, A084140, A084141.
%K A084142 nonn
%O A084142 1,2
%A A084142 _Harry J. Smith_, May 15 2003
%E A084142 Name edited by _Jon E. Schoenfield_, Oct 08 2023