Template:Sequence of the Day for September 13

Intended for: September 13, 2011

Timetable

• First draft entered by Charles R Greathouse IV on July 13, 2011 ✓
• Draft reviewed by Alonso del Arte on July 13, 2011 ✓
• Draft approved by Alonso del Arte on August 11, 2011 ✓

Yesterday's SOTD * Tomorrow's SOTD

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A070243:

Card{ k | φ (k)   ≤   n }, n   ≥   1

.

{ 2, 5, 5, 9, 9, 13, 13, 18, 18, 20, 20, 26, 26, 26, 26, 32, 32, 36, 36, 41, 41, 43, 43, 53, ... }

This sequence counts the number of positive integers

k

such that

φ (k)   ≤   n

, where

φ (k)

is Euler’s totient function. It has natural density

ζ  (2) ζ  (3) ζ  (6 )   ≈   1.9435

, so in some sense the values of

φ (k)

are on average not much less than

k

. Note that

a (n)  −  a (n  −  1) = Card{ k | φ (k) = n }, n   ≥   1,

(see A014197) and thus

a (n) = ∑ n m  = 1 A014197(m)

.