OFFSET
1,2
COMMENTS
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).
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).
In the range 1 .. 2^31 there are 782 such numbers, of which only the initial 1 is odd.
LINKS
PROG
(PARI)
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
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); (gcd(t-n, u)==u); };
\\ Alternatively as:
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); ((u>0)&&(0==((t-n)%u))); };
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, May 28 2021
STATUS
approved