A266047 Smallest integers of each prime signature of prime factorization palindromes (A265640).

1, 2, 4, 8, 12, 16, 32, 36, 48, 64, 72, 128, 144, 180, 192, 256, 288, 432, 512, 576, 720, 768, 900, 1024, 1152, 1296, 1728, 1800, 2048, 2304, 2592, 2880, 3072, 3600, 4096, 4608, 5184, 6300, 6480, 6912, 7200, 8192, 9216, 10368, 10800, 11520, 12288, 14400, 15552, 16384, 18432

Offset: 1

Author

Vladimir Shevelev, Dec 20 2015

Keywords

• nonn

Comments

A subsequence of A025487.

According to Hardy and Ramanujan, the number Q(x) of numbers

2^b_2*3^b_3*...*p^b_p <= x, (1)

where b_2>=b_3>=...>=b_p, is of order e^(2Pi/sqrt(3)(1+o(1))sqrt(log x/loglog x)).

If all b_i=2*c_i are even, then the number of such numbers is Q(sqrt(x)). Note that, if in (1) c_p>0, where p is n-th prime, then c_r>0, r=2 [Dusart], Eq(4.2),

p<=e*n*log(n)

Let K(x) be the number of a(n)<=x, q=nextprime(p). Then K(x)<=Q(sqrt(x))(1+Sum_{prime p}1/p)+1/3, where p satisfies (2) (+1/3, taking into account 1/q).

By [Rosser], Sum_{p<=x}1/p=loglog(x)+0.261497...+o(1). Hence K(x)<=Q(sqrt(x))*(loglog(e/2*log(x*loglogx))+1.594830...+o(1)).

Asymptotics of K(x) remain open.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• P. Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442 [math.NT], 2010.
• G. H. Hardy and S. Ramanujan, Asymptotic formulas concerning the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132.
• J. B. Rosser. Explicit bounds for some functions of prime numbers. Amer. J. Math. 63 (1941), 211-232.

Crossrefs

Cf. A025487, A265640, A265641.

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