A178139 -id:A178139 - cp's OEIS frontend

A377320 a(n) is the smallest positive integer k such that n + k and n - k have the same number of prime factors.

1, 1, 1, 3, 2, 2, 2, 6, 1, 5, 3, 2, 3, 6, 1, 1, 3, 2, 9, 2, 2, 5, 3, 4, 6, 1, 1, 11, 6, 4, 1, 6, 2, 2, 2, 2, 3, 8, 1, 1, 3, 2, 4, 3, 4, 12, 1, 1, 3, 2, 3, 1, 1, 3, 2, 7, 1, 4, 7, 4, 3, 6, 5, 1, 2, 1, 3, 5, 1, 3, 4, 4, 3, 1, 4, 13, 6, 2, 5, 15, 2, 7, 1, 3, 3, 1, 3

Offset: 4

Felix Huber, Nov 17 2024

nonn

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

			a(7) = 3 because 10 and 4 have both two prime factors. 8 and 6 or 9 and 7 respectively have a different number of prime factors.

Felix Huber, Table of n, a(n) for n = 4..10000

Cf. A001222, A082467, A178139, A377319, A377321.

A377320:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-Omega(n+k)=NumberTheory:-Omega(n-k) then
         return k
      fi
   od;
end proc;
seq(A377320(n),n=4..90);

A377320[n_] := Module[{k = 0}, While[PrimeOmega[++k + n] != PrimeOmega[n - k]]; k];
Array[A377320, 100, 4] (* Paolo Xausa, Dec 02 2024 *)

a(n) = my(k=1); while (bigomega(n+k) != bigomega(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

1 <= a(n) <= A082467(n).

A188348 Distance between n and the next number with the same number of distinct prime factors.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 2, 4, 1, 1, 2, 2, 2, 2, 2, 2, 5, 2, 12, 1, 5, 1, 1, 1, 2, 4, 1, 1, 4, 2, 18, 4, 1, 1, 2, 2, 2, 4, 1, 1, 2, 6, 1, 1, 1, 1, 4, 2, 6, 3, 1, 2, 3, 3, 4, 4, 1, 3, 8, 2, 2, 6, 1, 1, 1, 3, 6, 2, 2, 2, 3, 6, 6, 1, 1, 1, 3, 8, 12

Offset: 2

Juri-Stepan Gerasimov, Mar 28 2011

nonn

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A001221, A178139.

N:= 200: # to use numbers up to N
V:= Array(2..N, n -> nops(numtheory:-factorset(n))):
A:= Array(2..N):
for n from 2 to N do
  v:= V[n];
  if assigned(Last[v]) then
     A[Last[v]]:= n-Last[v]
  fi;
  Last[v]:= n;
od:
m:= min(select(t -> A[t]=0, [$2..N]))-1:
seq(A[i],i=2..m); # Robert Israel, Jan 24 2020

for(n=2, 100, f=omega(n); m=n+1; while(omega(m)!=f, m++); print1(m-n, ", ")) \\ Colin Barker, Jun 27 2014

{min d>0: A001221(n+d) = A001221(n)}.

Renamed, edited and checked by R. J. Mathar, Mar 28 2011.

cp's OEIS Frontend

A377320 a(n) is the smallest positive integer k such that n + k and n - k have the same number of prime factors.

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