A394062 - OEIS

%I #28 Mar 16 2026 15:58:44

%S 2,3,7,13,13,19,31,31,43,43,67,73,103,139,157,163,193,193,211,229,211,

%T 283,313,379,367,439,433,409,541,571,613,643,613,691,571,757,823,829,

%U 859,883,967,1033,1069,1063,1063,1237,1291,1259,1039,1399,1399,1453,1453,1531,1663,1669

%N a(n) = A393096(n) - sopf(A393096(n)) where sopf(n) is the sum of distinct primes of n.

%C All terms are prime.

%F a(n) = A393096(n)-A008472(A393096(n)).

%e For n=8, a(8) = 31 which added to 13 gives 44.  The distinct primes of 44 are 2 and 11 which sum to 13.

%t sopf[k_]:=Total[First/@FactorInteger[k]];q[k_]:=PrimeQ[sopf[k]]&&PrimeQ[k-sopf[k]];Select[Range[1723],q]-sopf/@Select[Range[1723],q] (* _James C. McMahon_, Mar 16 2026 *)

%o (Python)

%o from sympy import isprime, primefactors

%o def ok(n):return isprime(sum(primefactors(n))) and isprime(n-sum(primefactors(n)))

%o print(list(map(lambda n: n-sum(primefactors(n)), filter(ok, range(1, 2000)))))

%o (PARI) sopf(k) = vecsum(factor(k)[,1]); \\ A008472

%o isok(k) = my(s=sopf(k)); isprime(s) && isprime(k-s); \\ A393096

%o apply(x->x-sopf(x), select(isok, [1..1000])) \\ _Michel Marcus_, Mar 09 2026

%Y Cf. A008472, A394061, A393096.

%K nonn

%O 1,1

%A _Guy Siviour_, Mar 09 2026