A279935 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.

3, 4, 75, 112, 2057, 9178, 29818, 73813, 138992, 240469, 531002, 661489, 716856, 763648, 905474, 1033909, 1395554, 1572001, 1605519, 1643372, 1661030, 1692277, 1705724, 2312593, 2864773, 2911839, 2928193, 2977676, 3114366, 3744951, 4035647, 4122178, 4227036, 5716177

Offset: 1

Author

Paolo P. Lava, Dec 23 2016

Keywords

• nonn
• easy

Examples

			Prime factors of 9178 are 2, 13, 353:
sopf(9178) = 2 + 13 + 353 = 368, rad(9178) = 2 * 13 * 353 = 9178 and 9178 + 368 + 9178 = 18724.
Prime factors of 18724 are 2, 2, 31, 151:
sopf(18724) = 2 + 31 + 151 = 184, rad(18724) = 2 * 31 * 151 = 9362 and 18724 - 184 - 9362 = 9178.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..145

Crossrefs

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

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 1 to q do
a:=ifactors(n)[2]; b:=mul(a[k][1],k=1..nops(a))+add(a[k][1],k=1..nops(a));
c:=n+b; a:=ifactors(c)[2]; b:=mul(a[k][1],k=1..nops(a))+add(a[k][1],k=1..nops(a));
d:=c-b; if d=n then print(n); fi; od; end: P(10^9);

Mathematica

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 *)