cp's OEIS Frontend

A101593 a(n) is the number of m <= 2^n which are in A075190, i.e., such that m^2 is exactly at the center between two consecutive primes, or in other words A056929(m) = 0.

%I A101593 #55 Dec 14 2022 16:25:46
%S A101593 1,2,3,6,9,14,19,34,62,109,202,336,587,1100,2003,3630,6784,12607,
%T A101593 23647,44206,83510,157851,299810,571264,1090986,2088445,4004347,
%U A101593 7687694,14788984,28496850,54955214,106159961
%N A101593 a(n) is the number of m <= 2^n which are in A075190, i.e., such that m^2 is exactly at the center between two consecutive primes, or in other words A056929(m) = 0.
%C A101593 A056929(m) = 0 iff m^2 is an interprime <=> m^2 = (A007491(m^2) + A053001(m^2))/2 = average of the next higher and next lower primes.
%C A101593 From _M. F. Hasler_, Oct 18 2022: (Start)
%C A101593 The ratio a(n+1)/a(n) oscillates between 1.5 and 2 for the first few values, but then seems to converge to a limit between 1.9 and 2; from n = 19 on these ratios appear to be strictly increasing (from 1.87 at n = 19 to 1.92 at n = 27).
%C A101593 At first sight it seems natural that there are twice as many interprimes of the form f(m) when the upper limit on m is increased by a factor of 2, but this might depend on the function f.
%C A101593 If instead of m^2 we consider the same for m^3, then we find the sequence 0, 1, 1, 3, 5, 8, 18, 29, 52, 86, 136, 223, 421, 758, 1376, 2517, 4616, 8714, 16173, 30414, 57583, 109539, ... which follows roughly the same asymptotic behavior. (End)
%F A101593 a(1) counts the squares m^2 with m <= 2^n = 2 which are interprimes. The squares 0^2 = 0 and 1^2 = 1 obviously aren't interprimes, so the only such square in that range is m^2 = 2^2 = 4 = (nextprime + precprime)/2 = (3 + 5)/2, so a(1) = 1.
%F A101593 Then for n = 2, up to m <= 2^n = 4 we have the additional squares m^2 = 3^2 = 9 = (7 + 11)/2 (an interprime) and m^2 = 4^2 = 16 <> (13 + 17)/2 = 15, so this m^2 is not an interprime, and a(2) = a(1) + 1 = 2.
%o A101593 (PARI) a(n)=sum(i=2,2^n,!A056929(i))
%o A101593 (PARI) a(n)=sum(i=2,2^n,nextprime(i^2)+precprime(i^2)==2*i^2)
%o A101593 (PARI)
%o A101593 t=0;vector(15,n,t+=sum(i=1/2<<n+1,1<<n,nextprime(i^2)+precprime(i^2)==2*i^2))
%o A101593 /* Then, to compute progressively more values: (uses the variable t set to a(15) by the previous command; to continue the computation, set t to the last value printed and restart at the next index. [Explanation added by _M. F. Hasler_, Oct 18 2022] */
%o A101593 for(n=16, 30, print1("/* a("n") = */ ", t += sum(i=2^(n-1)+1,2^n, nextprime(i^2)+precprime(i^2)==2*i^2),", "))
%Y A101593 Cf. A007491, A053001, A056929, A069495, A075190.
%K A101593 nonn,more
%O A101593 1,2
%A A101593 _Zak Seidov_ and _M. F. Hasler_, Dec 27 2007
%E A101593 a(23)-a(25) from _Kevin P. Thompson_, Nov 26 2021
%E A101593 a(26)-a(28) from _M. F. Hasler_, Oct 18 2022
%E A101593 a(29)-a(32) from _Bill McEachen_, Dec 14 2022