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 A229978 #19 Jun 02 2025 08:40:21 %S A229978 7,22,31,37,52,67,82,94,97,112,115,127,136,142,148,157,172,178,187, %T A229978 199,202,214,217,220,232,241,247,262,277,283,292,304,307,322,325,337, %U A229978 346,352,367,382,388,397,409,412,427,430,442,445,451,457,472,487,502,517,532,535,547,562,577,592,598,607,622,637,643,652,661,667,682,697,712,724,727,742,757,772,787,802,808,817 %N A229978 Numbers k such that (2*k+1) + phi(2*k+1) <= sigma(2*k+1). %C A229978 It appears that the equation x + phi(x) = sigma(x) has the unique solution x=2. It is easy to show that this is the only even solution to the equation, but for odd solutions this is less obvious. The present sequence is motivated by the observation that for most odd numbers, the l.h.s. is larger than the r.h.s. (while the opposite is the case for all even numbers). (See also formulas in A228947.) %C A229978 From _Amiram Eldar_, Dec 23 2024: (Start) %C A229978 If k is an odd abundant number (A005231), then (k-1)/2 is a term of this sequence. %C A229978 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 9, 99, 981, 9879, 98613, 984293, 9850470, 98496984, 985005850, 9850433480, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0985... . (End) %H A229978 Amiram Eldar, <a href="/A229978/b229978.txt">Table of n, a(n) for n = 1..10000</a> %t A229978 Select[Range[1000], DivisorSigma[1, 2*#+1] > EulerPhi[2*#+1] + 2*#+1 &] (* _Amiram Eldar_, Dec 23 2024 *) %o A229978 (PARI) select(n->(2*n+1+eulerphi(2*n+1)<sigma(2*n+1)),vector(900,n,n-1)) %Y A229978 Cf. A000010, A000203, A005231, A051612 and references there, A228947. %K A229978 nonn,easy %O A229978 1,1 %A A229978 _M. F. Hasler_, Oct 05 2013