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Numbers k such that 1/(1/phi(k) + 1/phi(k+1)) is an integer.
6

%I #23 Feb 18 2021 14:16:41

%S 3,12,13,15,35,36,37,55,61,72,73,86,90,96,99,104,108,119,126,154,157,

%T 164,175,182,192,193,194,244,255,277,286,296,304,313,352,362,364,369,

%U 378,397,421,432,455,457,483,495,515,527,541,545,560,576,584,602,609

%N Numbers k such that 1/(1/phi(k) + 1/phi(k+1)) is an integer.

%H Amiram Eldar, <a href="/A073542/b073542.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A073542/a073542.jpg">Plot of a(n)/n^(3/2) and a(n)/(n*log(n)) for n = 1..100000</a>

%F Is a(n) asymptotic to c*n^(3/2) with 1<c<1.5? [This conjecture is false, see plots. - _Vaclav Kotesovec_, Feb 15 2019]

%e 1/phi(286) + 1/phi(287) = 1/120 + 1/240 = 1/80 so 286 is in the sequence.

%t Select[Range[700], IntegerQ[1/(1/EulerPhi[ # ]+1/EulerPhi[ #+1])]&]

%o (PARI) isok(k) = numerator(1/eulerphi(k) + 1/eulerphi(k+1)) == 1; \\ _Michel Marcus_, Feb 18 2021

%Y Cf. A000010, A073543, A073544.

%K easy,nonn

%O 1,1

%A _Benoit Cloitre_, Aug 27 2002

%E Edited by _Dean Hickerson_, Aug 31 2002