A307962 - OEIS

%I #16 Dec 02 2019 07:06:21

%S 1718200,4818880,5154600,12027400,14456640,22336600,29209400,32645800,

%T 33732160,36082200,39518600,49827800,53264200,62645440,63573400,

%U 67009800,70446200,73882600,80755400,81920960,87628200,91064600,91558720,97937400,101196480,101373800

%N Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

%C The larger counterparts are in A307963.

%C If (m, n) is an amicable pair (A259180), then the pair (m*k, n*k) with k=rad(m*n) is a coreful amicable pair (rad(i)=A007947(i) is the squarefree kernel of i), and so are all the pairs (m*k*s, n*k*s) where s is a squarefree number with gcd(s, k) = 1. Proof: k = rad(m*n) = rad(m)*rad(n)/rad(gcd(m,n)), csigma(m*k) = csigma(m*rad(m)*j) where j = rad(n)/rad(gcd(m,n)) is squarefree and coprime to m*rad(m), so csigma(m*k) = j * csigma(m*rad(m)) = j * rad(m)* sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * (n+m) = k *(n+m) = csigma(n*k).

%H Amiram Eldar, <a href="/A307962/b307962.txt">Table of n, a(n) for n = 1..10000</a>

%t f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[m = csigma[n] - n; If[m > n && csigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^8}]; s

%Y Cf. A007947, A057723, A063990, A259180, A307958, A307959, A307963.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 08 2019

%E Wrong terms corrected by _Amiram Eldar_, Dec 02 2019