A065550 a(n) = floor(sqrt(phi(w)*sigma(w)+w^2)), where w=10^n.

13, 136, 1391, 14030, 140865, 1411444, 14128309, 141352267, 1413868217, 14140409111, 141412724154, 1414170403052, 14141919829640, 141420277272713, 1414208167563878, 14142108649717545, 141421221367320690, 1414212888023339560, 14142132251982630599, 141421339378569021517

Offset: 1

Author

Labos Elemer, Nov 13 2001

Keywords

• nonn

Comments

a(n) tends to sqrt(2)*(10^n) when n->oo.

Links

• Robert Israel, Table of n, a(n) for n = 1..995

Crossrefs

Cf. A000290, A062354, A065655, A065656.

Programs

Maple

a:= n -> floor(sqrt(2*100^n - 20^n/5 - 50^n/2 + 10^n/10)):
map(a, [$1..100]); # Robert Israel, Dec 03 2024

Mathematica

a[n_] := Floor[Sqrt[EulerPhi[10^n] * DivisorSigma[1, 10^n] + 100^n]]; Array[a, 20] (* Amiram Eldar, Jun 12 2022 *)

PARI

a(n) = my(w=10^n); sqrtint(eulerphi(w)*sigma(w)+w^2); \\ Michel Marcus, Mar 23 2020

Python

from sympy import integer_nthroot, totient as phi, divisor_sigma as sigma
def isqrt(n): return integer_nthroot(n, 2)[0]
def a(n): w = 10**n; return isqrt(phi(w)*sigma(w, 1) + w**2)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 12 2022

Formula

a(n) = floor(sqrt(A062354(w) + A000290(w))), where w=10^n.

a(n) = floor(10^n * sqrt(2 - 5^(-n-1) - 2^(-n-1) + 10^(-n-1))). - Robert Israel, Dec 03 2024

Extensions

Corrected and extended by Michel Marcus, Jun 12 2022