A175760 -id:A175760 - cp's OEIS frontend

A207006 Numbers n such that omega(n) = omega(n + omega(n)) where omega(n) is the number of distinct primes dividing n.

1, 2, 3, 4, 7, 8, 10, 12, 16, 18, 20, 22, 24, 26, 31, 33, 34, 36, 38, 44, 46, 48, 50, 52, 54, 55, 56, 63, 72, 74, 75, 80, 85, 86, 91, 92, 93, 94, 96, 98, 102, 104, 106, 115, 116, 117, 122, 127, 133, 134, 141, 142, 143, 144, 145, 146, 153, 158, 159, 160, 162

Offset: 1

Michel Lagneau, Feb 14 2012

nonn

omega is the function in A001221. If there are infinitely many Sophie Germain primes (see A005384), then this sequence is infinite. Proof : the numbers of the form 4p are in a subsequence if p and 2p+1 are both primes, because from the property that omega(4p) = 2 and omega (p(2p+1)) = 2, if n = 4p then omega (n+omega(n)) = omega (4p + 2) = omega (2(2p+1)) = 2 = omega (n).

			12 is in the sequence because omega(12) = 2, omega(12 + 2) = omega(14) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001221, A207005, A005384 , A175760, A175759.

Select[Range[5*10^2],PrimeNu[#]==PrimeNu[#+PrimeNu[#]]&]

is(n)=my(o=omega(n));o==omega(n+o) \\ Charles R Greathouse IV, Feb 14 2012

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.

Original entry on oeis.org

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

Paolo P. Lava, Dec 23 2016

			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.

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

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

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);

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 *)

cp's OEIS Frontend

A207006 Numbers n such that omega(n) = omega(n + omega(n)) where omega(n) is the number of distinct primes dividing n.

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