A063070 a(n) = phi(n) - d(n), where d(n) is the number of divisors function (A000005).

0, -1, 0, -1, 2, -2, 4, 0, 3, 0, 8, -2, 10, 2, 4, 3, 14, 0, 16, 2, 8, 6, 20, 0, 17, 8, 14, 6, 26, 0, 28, 10, 16, 12, 20, 3, 34, 14, 20, 8, 38, 4, 40, 14, 18, 18, 44, 6, 39, 14, 28, 18, 50, 10, 36, 16, 32, 24, 56, 4, 58, 26, 30, 25, 44, 12, 64, 26, 40, 16, 68, 12, 70, 32, 34, 30, 56, 16, 76, 22, 49, 36, 80, 12, 60, 38, 52, 32, 86, 12, 68, 38

Offset: 1

Author

Jason Earls, Aug 04 2001

Keywords

• easy
• sign

Comments

It is known that a(n) >= 1 for n >= 31.

References

• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A000005, A000010 (phi). A020488 gives n such that a(n) = 0.

Programs

Mathematica

Table[EulerPhi[n] - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 24 2021 *)

PARI

j=[]; for(n=1,150,j=concat(j,eulerphi(n)-(numdiv(n)))); j

PARI

{ for (n=1, 1000, write("b063070.txt", n, " ", eulerphi(n) - numdiv(n)) ) } \\ Harry J. Smith, Aug 16 2009

Formula

a(n) = A000010(n) - A000005(n). - Wesley Ivan Hurt, Nov 24 2021

a(n) = Sum_{d|n} (d*mu(n/d) - 1). - Wesley Ivan Hurt, Jul 21 2025