A065656 - OEIS

%I #21 Jun 15 2018 08:06:36

%S 1169,7777,41111,46097,668167,846817,2107519,3612769,17424241,

%T 30666527,37526993,56323393,214746055,383523857,512376769,1021934641,

%U 1228492849,1303949599,4056001351,7425397169,17073544447,17859428369,18452226887,46874737969,51411954391

%N Composite numbers k such that sigma(k)*phi(k) + 2*(k+1) is a square.

%C a(n) and square root of phi(a(n))*sigma(a(n)) + 2*a(n) + 2 are close to each other: e.g., a(7) = 2107519 and this square root is 2107458.

%C Since (p+1)*(p-1) + 2*(p+1) = p*p + 2*p + 1 = (p+1)^2 is a square, all primes are solutions.

%C 73362272287 and 181264312447 are also terms. - _Donovan Johnson_, Jul 13 2012

%e k = 7777: sigma(7777) = 9792, phi(7777) = 6000 and 9792*6000 + 2*7778 = 587675556 = 7666^2.

%o (PARI) { n=0; for (m=1, 10^12, if (isprime(m), next); s=sigma(m)*eulerphi(m) + 2*(m + 1); if (issquare(s), write("b065656.txt", n++, " ", m); if (n==100, return)) ) } \\ _Harry J. Smith_, Oct 26 2009

%Y Cf. A062354, A000203, A000010, A065655.

%K nonn

%O 1,1

%A _Labos Elemer_, Nov 12 2001

%E a(9)-a(15) from _Harry J. Smith_, Oct 26 2009

%E a(16)-a(20) from _Donovan Johnson_, May 24 2011

%E a(21)-a(25) from _Donovan Johnson_, Jul 13 2012