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A084596 a(n) is the number of times n is in sequence A014085; i.e., there are exactly a(n) cases where there are exactly n primes between m^2 and (m+1)^2 for m >= 0.

%I A084596 #13 Feb 16 2025 08:32:49
%S A084596 1,0,4,3,4,4,4,5,4,7,5,5,5,7,2,7,5,6,4,4,5,10,6,9,7,5,2,5,6,6,10,4,5,
%T A084596 11,5,3,8,3,8,9,7,10,5,4,6,8,8,5,6,10,8,9,4,4,6,7,8,7,5,10,9,9,6,8,7,
%U A084596 7,7,8,6,3,5,8,4,8,14,8,7,9,10,6,9,6,7,6,6,8,10,4,8,7,6,8,5,14,6,7,11,7,10,8
%N A084596 a(n) is the number of times n is in sequence A014085; i.e., there are exactly a(n) cases where there are exactly n primes between m^2 and (m+1)^2 for m >= 0.
%C A084596 This sequences uses a finite number of terms of A014085 to conjecture the behavior of all terms of A014085. The first 10000 terms of this sequence were computed using 120000 terms of A014085. Legendre's conjecture is equivalent to a(0) remaining 1 for all terms of A014085. [Comment reworded by _T. D. Noe_, Sep 04 2008]
%D A084596 P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 143.
%H A084596 T. D. Noe, <a href="/A084596/b084596.txt">Table of n, a(n) for n = 0..10000</a>
%H A084596 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LandausProblems.html">Landau's Problems.</a>
%e A084596 a(14)=2 because 14 is in sequence A014085 only two times. There are 14 primes between 64^2 and 65^2 as well as between 77^2 and 78^2. These are the only cases with exactly 14 primes.
%Y A084596 Cf. A007491, A014085, A084597.
%K A084596 nonn
%O A084596 0,3
%A A084596 _Harry J. Smith_, May 31 2003