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A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

%I A344700 #29 May 29 2021 20:02:07
%S A344700 1,6,24,28,168,496,864,1080,1520,1836,2016,2088,2112,2520,2912,2976,
%T A344700 3000,3024,3240,3800,8128,9000,11088,11232,11448,14160,14688,16920,
%U A344700 17028,18360,19872,20520,20880,25280,25488,27552,29376,30800,31200,31320,31968,35400,39240,44064,48768,49896,50760,51480,51660,52200,55680
%N A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.
%C A344700 Numbers k for which A344704(k) = A344705(k), i.e., numbers k such that gcd(A001615(k)-k, A001615(k)-A001065(k)) = A001615(k) - A001065(k).
%C A344700 Note that A306927(k) is always nonnegative, but A344705(k) = A033879(k) + A306927(k) gets also negative values. Number k is perfect only when A033879(k) = A344705(k) - A306927(k) = 0, that is, when A344705(k) = A306927(k), which necessitates that A306927(k) should be a multiple of A344705(k), and their quotient should be nonnegative (actually = +1).
%C A344700 In the range 1 .. 2^31 there are 782 such numbers, of which only the initial 1 is odd.
%H A344700 Antti Karttunen, <a href="/A344700/b344700.txt">Table of n, a(n) for n = 1..782 (terms < 2^31)</a>
%H A344700 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%o A344700 (PARI)
%o A344700 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
%o A344700 isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); (gcd(t-n,u)==u); };
%o A344700 \\ Alternatively as:
%o A344700 isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); ((u>0)&&(0==((t-n)%u))); };
%Y A344700 Cf. A000203, A001065, A001615, A033879, A244963, A306927, A344704, A344705, A344752 (gives the quotient A306927(k)/A344705(k) computed for these terms), A344753.
%Y A344700 Cf. A000396 (subsequence).
%Y A344700 Cf. also A344754, A344755.
%K A344700 nonn
%O A344700 1,2
%A A344700 _Antti Karttunen_, May 28 2021