cp's OEIS Frontend

A372455 Smaller term of each Ruth-Aaron pair in which the sum of distinct prime factors is a prime number.

%I A372455 #60 Feb 16 2025 08:34:06
%S A372455 5,24,49,714,1682,12726,13775,25839,26642,75140,79118,95709,109939,
%T A372455 189080,197657,204258,228599,235586,268656,319428,384312,416119,
%U A372455 547525,554682,560150,563390,565823,576984,608316,740726,823150,839375,850746,851709,869054,890723,901747
%N A372455 Smaller term of each Ruth-Aaron pair in which the sum of distinct prime factors is a prime number.
%C A372455 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 A372455 The present sequence is those k for which this common sopf(k) = sopf(k+1) is prime.
%H A372455 Numberphile, <a href="https://www.youtube.com/watch?v=aCq04N9it8U">Aaron Numbers</a>.
%H A372455 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ruth-AaronPair.html">Ruth-Aaron Pair</a>.
%H A372455 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ruth%E2%80%93Aaron_pair">Ruth-Aaron pair</a>.
%e A372455 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 A372455 SumPF := n -> add(NumberTheory:-PrimeFactors(n)):
%p A372455 aList := proc(upto) local s0, s1, L, k; s0, s1 := 2, 3; L := NULL;
%p A372455 for k from 1 to upto do s0, s1 := s1, SumPF(k + 1);
%p A372455     if s0 = s1 then if isprime(s0) then L := L, k fi fi;
%p A372455 od; L end: aList(13000);  # _Peter Luschny_, Jun 11 2024
%t A372455 s[n_] := s[n] = Plus @@ FactorInteger[n][[;; , 1]]; Select[Range[10^6], PrimeQ[s[#]] && s[# + 1] == s[#] &] (* _Amiram Eldar_, May 11 2024 *)
%o A372455 (Python)
%o A372455 from sympy import isprime, primefactors
%o A372455 for k in range(10**6):
%o A372455     s0, s1 = sum(primefactors(k)), sum(primefactors(k + 1))
%o A372455     if s0 == s1 and isprime(s0): print(k, end=', ') # _Jason Yuen_, Jun 05 2024
%o A372455 (PARI) sopf(n) = my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1]); \\ A008472
%o A372455 isok(n) = my(s=sopf(n)); isprime(s) && (s==sopf(n+1)); \\ _Michel Marcus_, May 11 2024
%Y A372455 Subsequence of A006145.
%Y A372455 Cf. A008472.
%K A372455 nonn,easy
%O A372455 1,1
%A A372455 _Gerardo Salcido Martinez_, May 01 2024
%E A372455 More terms from _Michel Marcus_, May 11 2024