This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A011774 #55 Feb 16 2025 08:32:32
%S A011774 1,312,560,588,1400,23760,59400,85632,147492,153720,556160,569328,
%T A011774 1590816,2013216,3343776,4563000,4695456,9745728,12558912,22013952,
%U A011774 23336172,30002960,45326160,52021242,75007400,113315400,137617728,153587720,402831360,699117024
%N A011774 Nonprimes k that divide sigma(k) + phi(k).
%C A011774 2*k = sigma(k) + phi(k) if and only if k is 1 or a prime.
%C A011774 If 7*2^j - 1 is prime then m = 2^(j+2)*3*(7*2^j - 1) is in the sequence. Because phi(m) = 2^(j+2)*(7*2^j - 2); sigma(m) = 7*2^(j+2)*(2^(j+3) - 1) so phi(m) + sigma(m) = 2^(j+2)*((7*2^j - 2) + (7*2^(j+3) - 7)) = 2^(j+2)* (63*2^(j+2) - 9) = 3*(2^(j+2)*3*(7*2^j - 1)) = 3*m, hence m is a term of A011251 and consequently m is a term of this sequence. A112729 gives such m's. - _Farideh Firoozbakht_, Dec 01 2005
%C A011774 Conjecture: For n > 1, a(n) is a Zumkeller number (A083207). Verified for all n in [2,63]. - _Ivan N. Ianakiev_, Jan 25 2023
%D A011774 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
%D A011774 Zhang Ming-Zhi, typescript submitted to Unsolved Problems section of Monthly, 96-01-10.
%H A011774 Giovanni Resta, <a href="/A011774/b011774.txt">Table of n, a(n) for n = 1..63</a> (terms < 10^13; first 53 terms from Donovan Johnson)
%H A011774 Richard K. Guy, <a href="http://www.jstor.org/stable/2974586">Divisors and desires</a>, Amer. Math. Monthly, 104 (1997), 359-360.
%H A011774 Q.-X. Jin and M. Tang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Tang2/tang72.html">The 4-Nicol Numbers Having Five Different Prime Divisors</a>, J. Int. Seq. 14 (2011) # 11.7.2.
%H A011774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeNumber.html">Prime Number</a>.
%e A011774 a(26) = 113315400: sigma = 426535200, phi = 26726400, quotient = 4.
%t A011774 Do[If[Mod[DivisorSigma[1, n]+EulerPhi[n], n]==0, Print[n]], {n, 1, 2*10^7}]
%t A011774 Do[ If[ ! PrimeQ[n] && Mod[ DivisorSigma[1, n] + EulerPhi[n], n] == 0, Print[n] ], {n, 1, 10^8} ]
%o A011774 (PARI) sp(n)=my(f=factor(n));n*prod(i=1, #f[,1], 1-1/f[i,1]) + prod(i=1, #f[,1], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))
%o A011774 p=2;forprime(q=3, 1e6, for(n=p+1, q-1, if(sp(n)%n==0, print1(n", ")));p=q) \\ _Charles R Greathouse IV_, Mar 19 2012
%Y A011774 Cf. A065387, A011251, A011254, A055681, A001771, A112729.
%K A011774 nonn,nice
%O A011774 1,2
%A A011774 _R. K. Guy_
%E A011774 More terms from _David W. Wilson_
%E A011774 Corrected by _Labos Elemer_, Feb 12 2004