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A101593
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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.
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1
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1, 2, 3, 6, 9, 14, 19, 34, 62, 109, 202, 336, 587, 1100, 2003, 3630, 6784, 12607, 23647, 44206, 83510, 157851, 299810, 571264, 1090986, 2088445, 4004347, 7687694, 14788984, 28496850, 54955214, 106159961
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OFFSET
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1,2
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COMMENTS
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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.
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).
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.
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)
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LINKS
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FORMULA
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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.
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.
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PROG
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(PARI) a(n)=sum(i=2, 2^n, !A056929(i))
(PARI) a(n)=sum(i=2, 2^n, nextprime(i^2)+precprime(i^2)==2*i^2)
(PARI)
t=0; vector(15, n, t+=sum(i=1/2<<n+1, 1<<n, nextprime(i^2)+precprime(i^2)==2*i^2))
/* 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] */
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), ", "))
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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