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 A066192 #18 Apr 27 2025 09:08:41
%S A066192 4,8,12,16,24,32,48,56,64,96,112,120,128,192,224,240,256,384,448,480,
%T A066192 512,528,672,768,896,960,992,1024,1056,1344,1456,1536,1792,1920,1984,
%U A066192 2048,2112,2160,2208,2688,2912,3072,3584,3840,3968,4096,4224,4320,4416
%N A066192 Composite numbers k such that the sum of the odd aliquot parts of k divides k.
%C A066192 From _Amiram Eldar_, Apr 27 2025: (Start)
%C A066192 If there is an odd term in this sequence it must be an odd perfect number (A000396). If k is an odd term then d = sigma(k)-k divides k. If d < k then sigma(k) = k + d with d being an aliquot divisor of k which is possible only if k is prime and d = 1. Therefore, d = k and k must be an odd perfect number.
%C A066192 This sequence is infinite because if k is a term then 2*k is also a term. The primitive terms are in A383428. (End)
%H A066192 Amiram Eldar, <a href="/A066192/b066192.txt">Table of n, a(n) for n = 1..4905</a> (terms below 10^11; terms 1..1000 from Harry J. Smith)
%H A066192 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>.
%t A066192 Do[ d = Drop[ Divisors[ n ], -1 ]; l = Length[ d ]; od = 1; k = 2; While[ k <= l, If[ OddQ[ d[ [ k ] ] ], od = od + d[ [ k ] ] ]; k++ ]; If[ !PrimeQ[ n ] && IntegerQ[ n/od ], Print[ n ] ], {n, 2, 10^4} ]
%t A066192 Select[Range[4500], CompositeQ[#] && Divisible[#, DivisorSigma[1, #/2^IntegerExponent[#, 2]] - If[OddQ[#], #, 0]] &] (* _Amiram Eldar_, Apr 27 2025 *)
%o A066192 (PARI) { n=0; for (m=4, 10^9, if (isprime(m), next); d=divisors(m); s=1; for (i=2, numdiv(m) - 1, if (d[i]%2, s += d[i])); if (s > 0 && m%s == 0, write("b066192.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Feb 05 2010
%o A066192 (PARI) isok(k) = if(k == 1, 0, !isprime(k) && !(k % (sigma(k >> valuation(k, 2)) - if(k%2, k)))); \\ _Amiram Eldar_, Apr 27 2025
%Y A066192 Intersection of A002808 and A066191.
%Y A066192 Cf. A000396, A091570, A383428.
%K A066192 nonn
%O A066192 1,1
%A A066192 _Robert G. Wilson v_, Dec 15 2001