A372455 - OEIS

%I #59 Jun 11 2024 19:18:23

%S 5,24,49,714,1682,12726,13775,25839,26642,75140,79118,95709,109939,

%T 189080,197657,204258,228599,235586,268656,319428,384312,416119,

%U 547525,554682,560150,563390,565823,576984,608316,740726,823150,839375,850746,851709,869054,890723,901747

%N Smaller term of each Ruth-Aaron pair in which the sum of distinct prime factors is a prime number.

%C A Ruth-Aaron pair consists of two consecutive integers (k,k+1) such that sopf(k) = sopf(k+1) where sopf(x) is the sum of the distinct prime factors of x (A008472).

%C The present sequence is those k for which this common sopf(k) = sopf(k+1) is prime.

%H Numberphile, <a href="https://www.youtube.com/watch?v=aCq04N9it8U">Aaron Numbers</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ruth-AaronPair.html">Ruth-Aaron Pair</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ruth%E2%80%93Aaron_pair">Ruth-Aaron pair</a>.

%e 1682 is a term because the pair (1682, 1683) is a Ruth-Aaron pair with sum of prime factors 2 + 29 = 3 + 11 + 17 = 31 which is prime.

%p SumPF := n -> add(NumberTheory:-PrimeFactors(n)):

%p aList := proc(upto) local s0, s1, L, k; s0, s1 := 2, 3; L := NULL;

%p for k from 1 to upto do s0, s1 := s1, SumPF(k + 1);

%p     if s0 = s1 then if isprime(s0) then L := L, k fi fi;

%p od; L end: aList(13000);  # _Peter Luschny_, Jun 11 2024

%t s[n_] := s[n] = Plus @@ FactorInteger[n][[;; , 1]]; Select[Range[10^6], PrimeQ[s[#]] && s[# + 1] == s[#] &] (* _Amiram Eldar_, May 11 2024 *)

%o (Python)

%o from sympy import isprime, primefactors

%o for k in range(10**6):

%o     s0, s1 = sum(primefactors(k)), sum(primefactors(k + 1))

%o     if s0 == s1 and isprime(s0): print(k, end=', ') # _Jason Yuen_, Jun 05 2024

%o (PARI) sopf(n) = my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1]); \\ A008472

%o isok(n) = my(s=sopf(n)); isprime(s) && (s==sopf(n+1)); \\ _Michel Marcus_, May 11 2024

%Y Subsequence of A006145.

%Y Cf. A008472.

%K nonn,easy

%O 1,1

%A _Gerardo Salcido Martinez_, May 01 2024

%E More terms from _Michel Marcus_, May 11 2024