A244439 Numbers n such that phi(n)*sigma(n) = phi(n+1)*sigma(n+1).

5, 55, 56, 123, 135, 147, 175, 304, 351, 644, 1015, 2464, 19304, 61131, 162524, 476671, 567644, 712724, 801944, 2435488, 3346399, 3885056, 4555999, 8085560, 8369360, 12516692, 22702119, 29628800, 83884031, 83994624, 84789247, 354812535, 860616295, 1091535704

Offset: 1

Author

Farideh Firoozbakht, Aug 14 2014

Keywords

• nonn

Comments

Since both numbers 55 and 56 are in the sequence we have sigma(55)*phi(55) = sigma(56)*phi(56) = sigma(57)*phi(57). It seems that 56 is the only number n which has the nice property sigma(n-1)*phi(n-1) = sigma(n)*phi(n) = sigma(n+1)*phi(n+1).

Up to n < 6*10^11 the similar equation phi(n)*sigma(n+1) = phi(n+1)*sigma(n) is satisfied only by n = 696003. - Giovanni Resta, Jun 08 2020

Examples

			5 is in the sequence because sigma(5)*phi(5) = sigma(6)*phi(6) = 24.
55 is in the sequence because sigma(55)*phi(55) = sigma(56)*phi(56) = 2880.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..52 (terms < 10^13, first 40 terms from Jens Kruse Andersen)

Crossrefs

Cf. A000010, A000203, A145749.

Programs

Maple

with(numtheory): A244439:=n->`if`(phi(n)*sigma(n) = phi(n+1)*sigma(n+1), n, NULL): seq(A244439(n), n=1..10^4); # Wesley Ivan Hurt, Aug 16 2014

Mathematica

Select[Range[10^5], Equal @@ (EulerPhi[{#, # + 1}] DivisorSigma[1, {#, # + 1}]) &] (* Giovanni Resta, Jun 08 2020 *)

PARI

for(n=1,10^6,s=eulerphi(n)*sigma(n);if(s==eulerphi(n+1)*sigma(n+1),print1(n,", "))) \\ Derek Orr, Aug 14 2014

Extensions

More terms from Jens Kruse Andersen, Aug 16 2014