A065395 Commutator of sigma and phi functions.

0, -1, 1, -3, 5, -1, 8, -1, 0, 1, 14, -5, 22, 4, 7, -15, 25, -12, 31, 3, 12, 6, 28, -1, 12, 16, 23, 4, 48, -9, 56, -5, 26, 13, 44, -44, 73, 23, 36, 7, 78, -4, 76, 18, 36, 12, 56, -29, 60, -18, 39, 18, 80, 7, 66, 28, 59, 32, 74, -17, 138, 40, 43, -63, 100, -6

Offset: 1

Author

Labos Elemer, Nov 05 2001

Keywords

• sign
• easy

Comments

Golomb (1993) proved that the terms are both positive and negative infinitely often. - Amiram Eldar, Feb 27 2024

Examples

			n = 13: sigma(13) = 14, phi(14) = 6, phi(13) = 12, sigma(12) = 28, a(13) = 28-6 = 22.
		

References

• Solomon W. Golomb, Equality among number-theoretic functions, Abstracts Amer. Math. Soc., Vol. 14 (1993), pp. 415-416.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.
• Solomon W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)

Crossrefs

Cf. A000010, A000203, A033632 (positions of 0's), A062401, A062402.

Programs

Magma

[DivisorSigma(1, EulerPhi(n))-EulerPhi(DivisorSigma(1, n)): n in [1..70]]; // Bruno Berselli, Oct 20 2015

Maple

with(numtheory); A065395:=n->sigma(phi(n))-phi(sigma(n)); seq(A065395(n), n=1..100); # Wesley Ivan Hurt, Dec 26 2013

Mathematica

Table[DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]], {n, 100}] (* T. D. Noe, Nov 04 2013 *)

PARI

a(n) = { sigma(eulerphi(n)) - eulerphi(sigma(n)) } \\ Harry J. Smith, Oct 18 2009

Formula

a(n) = sigma(phi(n)) - phi(sigma(n)) = A000203(A000010(n)) - A000010(A000203(n)).

a(n) = A062402(n) - A062401(n). - Amiram Eldar, Feb 27 2024