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

1,2

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<p. Thus 2*3*...*p_n <= 2^c_2* ... p^c_p <= sqrt(x). By the PNT, 2*3*...*p_n=e^(n+o(n)). Then n<=log(x)/2(1+o(log(x))) and for n>=2 [Dusart], Eq(4.2),

p<=e*n*log(n)<e/2*log(x*loglogx). (2)

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.

Sequence in context: A326807 A052184 A152768 * A324214 A371732 A368507

Adjacent sequences: A266044 A266045 A266046 * A266048 A266049 A266050

Keyword

nonn

Author

Vladimir Shevelev, Dec 20 2015

Status

approved