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 A335369 #12 Jun 06 2020 03:18:58
%S A335369 1,6,140,496,672,2970,27846,105664,173600,237510,539400,695520,726180,
%T A335369 753480,1421280,1539720,2229500,2290260,8872200,11981970,14303520,
%U A335369 15495480,33550336,50401728,71253000,80832960,90409410,144963000,221557248,233103780,287425800,318177800
%N A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k.
%C A335369 If k is a harmonic number (A001599) and p is a prime that does not divide k, then k*p is a harmonic number if and only if (p+1)/2 is a divisor of the harmonic mean of the divisors of k, h(k) = k*tau(k)/sigma(k) = k*A000005(k)/A000203(k). The terms of this sequence are harmonic numbers k such that for all the divisors d of h(k), 2*d - 1 is either a nonprime or a prime divisor of k.
%C A335369 The even perfect numbers, 2^(p-1)*(2^p - 1) where p is a Mersenne exponent (A000043), have harmonic mean of divisors p. Therefore, they are in this sequence if p = 2 or if 2*p - 1 is composite (i.e., not in A172461). Of the first 47 Mersenne exponents there are 37 such primes (p = 2, 5, 13, 17, ...), with the corresponding even perfect numbers 6, 496, 33550336, 8589869056, ...
%H A335369 Amiram Eldar, <a href="/A335369/b335369.txt">Table of n, a(n) for n = 1..246</a> (terms below 10^14)
%H A335369 Mariano Garcia, <a href="https://www.jstor.org/stable/2307792">On numbers with integral harmonic mean</a>, The American Mathematical Monthly, Vol. 61, No. 2 (1954), pp. 89-96. See page 95.
%e A335369 1 is a term since it is a harmonic number, and there is no prime p such that 1*p = p is a harmonic number (if p is a prime, h(p) = 2*p/(p+1) cannot be an integer).
%t A335369 harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {_, _}][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; primeCountQ[n_] := Module[{d = Divisors[harMean[n]]}, Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &] == {}]; Select[harmNums, primeCountQ]
%Y A335369 Cf. A000005, A000043, A000203, A000396, A001599, A099377, A099378, A172461, A335368, A335370, A335371.
%K A335369 nonn
%O A335369 1,2
%A A335369 _Amiram Eldar_, Jun 03 2020