login
Numbers n such that n + sopf(n) + rad(n) = m and m - sopf(m) - rad(m) = n, where sopf(n) is the sum of the distinct primes dividing n and rad(n) is the squarefree kernel of n.
1

%I #11 Dec 24 2016 11:20:57

%S 3,4,75,112,2057,9178,29818,73813,138992,240469,531002,661489,716856,

%T 763648,905474,1033909,1395554,1572001,1605519,1643372,1661030,

%U 1692277,1705724,2312593,2864773,2911839,2928193,2977676,3114366,3744951,4035647,4122178,4227036,5716177

%N Numbers n such that n + sopf(n) + rad(n) = m and m - sopf(m) - rad(m) = n, where sopf(n) is the sum of the distinct primes dividing n and rad(n) is the squarefree kernel of n.

%H Robert G. Wilson v, <a href="/A279935/b279935.txt">Table of n, a(n) for n = 1..145</a>

%e Prime factors of 9178 are 2, 13, 353:

%e sopf(9178) = 2 + 13 + 353 = 368, rad(9178) = 2 * 13 * 353 = 9178 and 9178 + 368 + 9178 = 18724.

%e Prime factors of 18724 are 2, 2, 31, 151:

%e sopf(18724) = 2 + 31 + 151 = 184, rad(18724) = 2 * 31 * 151 = 9362 and 18724 - 184 - 9362 = 9178.

%p with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 1 to q do

%p a:=ifactors(n)[2]; b:=mul(a[k][1],k=1..nops(a))+add(a[k][1],k=1..nops(a));

%p c:=n+b; a:=ifactors(c)[2]; b:=mul(a[k][1],k=1..nops(a))+add(a[k][1],k=1..nops(a));

%p d:=c-b; if d=n then print(n); fi; od; end: P(10^9);

%t f[n_] := Block[{pd = First@# & /@ FactorInteger@n}, Times @@ pd + Plus @@ pd]; fQ[n_] := n + f[n] - f[n + f[n]] == n; Select[ Range@ 1000000, fQ] (* _Robert G. Wilson v_, Dec 24 2016 *)

%Y Cf. A008472, A007947, A050780, A050781, A175760.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Dec 23 2016