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 A385462 #18 Jul 06 2025 10:40:11
%S A385462 2,4,8,10,16,24,32,44,60,64,84,128,136,152,168,184,252,256,270,336,
%T A385462 512,630,752,756,792,864,884,924,936,1024,1140,1170,1488,1638,2048,
%U A385462 2144,2268,2272,2528,2808,2970,3672,4096,4320,4464,4680,5148,5472,6804,7308,7644,8192,8384
%N A385462 Numbers t which have a proper divisor d_i(t) such that (d_i(t) + sigma(t))/t is an integer k.
%C A385462 Consecutive elements of this sequence for which k = 2 are consecutive deficient-perfect numbers (A271816) > 1.
%C A385462 Consecutive elements of this sequence for which k = 3 are consecutive non-perfect elements of A364977.
%C A385462 Let b_k(m) be the number of elements of this sequence with the same k and <= m.
%C A385462 --------------------------------------------
%C A385462 m | b_2(m) | b_3(m) | b_4(m) | b_5(m) |
%C A385462 --------------------------------------------
%C A385462 10^3 | 16 | 13 | - | - |
%C A385462 10^4 | 24 | 31 | 2 | - |
%C A385462 10^5 | 37 | 62 | 5 | - |
%C A385462 10^6 | 54 | 107 | 19 | - |
%C A385462 10^7 | 73 | 175 | 43 | 1 |
%C A385462 10^8 | 98 | 254 | 80 | 3 |
%C A385462 10^9 | 128 | 357 | 141 | 13 |
%C A385462 --------------------------------------------
%C A385462 Are there any odd terms in this sequence for which k > 2? If they exist, they are > 10^9.
%C A385462 Contains 2^k * (2^(k+1) + 2^j - 1) if 0 <= j <= k and 2^(k+1) + 2^j - 1 is prime. - _Robert Israel_, Jun 30 2025
%e A385462 4 is in this sequence because sigma(4) + d_1(4) = 7 + 1 = 8 and 8/4 = 2.
%e A385462 24 is in this sequence because sigma(24) + d_7(24) = 60 + 12 = 72 and 72/24 = 3.
%e A385462 4320 is in this sequence because sigma(4320) + d_47(4320) = 15120 + 2160 = 17280 and 17280/4320 = 4.
%p A385462 filter:= proc(n) local s;
%p A385462 s:= - numtheory:-sigma(n) mod n;
%p A385462 ormap(d -> d mod n = s, numtheory:-divisors(n) minus {n})
%p A385462 end proc:
%p A385462 select(filter, [$1..10^4]); # _Robert Israel_, Jun 30 2025
%t A385462 Select[Range[8384],AnyTrue[(Drop[Divisors[#],-1]+DivisorSigma[1,#])/#,IntegerQ]&] (* _James C. McMahon_, Jul 05 2025 *)
%o A385462 (Maxima)
%o A385462 (n:1, for t:1 thru 10000 do (s:divsum(t), (A:args(divisors(t)),
%o A385462 for i:1 thru length(A)-1 do (y:s+A[i],
%o A385462 if mod(y,t)=0 then (print(n,"",t), n:n+1)))));
%o A385462 (PARI) isok(t) = my(s=sigma(t)); fordiv(t, d, if ((d<t) && (denominator((d+s)/t) == 1), return(1))); \\ _Michel Marcus_, Jun 30 2025
%Y A385462 Cf. A000005, A000203, A007691, A054027, A271816, A364977.
%K A385462 nonn
%O A385462 1,1
%A A385462 _Lechoslaw Ratajczak_, Jun 29 2025