A393096 Numbers k such that sopf(k) and k-sopf(k) are prime, where sopf() is the sum of distinct prime factors.

4, 10, 12, 18, 20, 24, 36, 44, 48, 50, 72, 80, 108, 144, 162, 176, 200, 210, 216, 242, 284, 288, 320, 384, 390, 462, 464, 548, 570, 576, 630, 648, 656, 704, 764, 780, 840, 858, 864, 944, 972, 1050, 1088, 1110, 1136, 1290, 1296, 1302, 1388, 1428, 1430, 1458, 1470, 1536, 1680, 1722, 1724, 1728, 1794, 1830, 1844, 1890, 2000

Offset

1,1

Comments

k is always even. If k is odd then sopf(k) is either even which cannot be a prime, or sopf(k) is odd so k-sopf(k) has to be even.

omega(a(n)) is even for n >= 2 where omega is A001221. - David A. Corneth, Mar 10 2026

Links

James C. McMahon, Table of n, a(n) for n = 1..10000

Example

For n=8, a(8) = 44, the distinct prime factors are 2 and 11, these sum to the prime number 13.  When this is added to the prime number 31 it returns the original number of 44.

Maple

filter:= proc(n) local v;
  v:= convert(NumberTheory:-PrimeFactors(n), `+`); isprime(v) and isprime(n-v)
end proc:
select(filter, [seq(i, i=2..3000, 2)]); # Robert Israel, Mar 12 2026

Mathematica

sopf[k_]:=Total[First/@FactorInteger[k]]; q[k_]:=PrimeQ[sopf[k]]&&PrimeQ[k-sopf[k]]; Select[Range[2000], q] (* James C. McMahon, Mar 16 2026 *)

Prog

(Python)
from sympy import isprime, primefactors
def ok(n):return isprime(sum(primefactors(n))) and isprime(n-sum(primefactors(n)))
print(list(filter(ok, range(1, 2000))))
(PARI) isok(k) = my(s=vecsum(factor(k)[, 1])); isprime(s) && isprime(k-s); \\ Michel Marcus, Mar 09 2026

Crossrefs

Cf. A001221, A008472, A394061, A394062.

Sequence in context: A109096 A079283 A190516 * A310341 A107960 A310342

Adjacent sequences: A393093 A393094 A393095 * A393097 A393098 A393099

Keyword

nonn

Author

Guy Siviour, Mar 09 2026

Status

approved