A372455 Smaller term of each Ruth-Aaron pair in which the sum of distinct prime factors is a prime number.
5, 24, 49, 714, 1682, 12726, 13775, 25839, 26642, 75140, 79118, 95709, 109939, 189080, 197657, 204258, 228599, 235586, 268656, 319428, 384312, 416119, 547525, 554682, 560150, 563390, 565823, 576984, 608316, 740726, 823150, 839375, 850746, 851709, 869054, 890723, 901747
Offset: 1
Examples
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.
Links
- Numberphile, Aaron Numbers.
- Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.
- Wikipedia, Ruth-Aaron pair.
Programs
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Maple
SumPF := n -> add(NumberTheory:-PrimeFactors(n)): aList := proc(upto) local s0, s1, L, k; s0, s1 := 2, 3; L := NULL; for k from 1 to upto do s0, s1 := s1, SumPF(k + 1); if s0 = s1 then if isprime(s0) then L := L, k fi fi; od; L end: aList(13000); # Peter Luschny, Jun 11 2024 -
Mathematica
s[n_] := s[n] = Plus @@ FactorInteger[n][[;; , 1]]; Select[Range[10^6], PrimeQ[s[#]] && s[# + 1] == s[#] &] (* Amiram Eldar, May 11 2024 *)
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PARI
sopf(n) = my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1]); \\ A008472 isok(n) = my(s=sopf(n)); isprime(s) && (s==sopf(n+1)); \\ Michel Marcus, May 11 2024
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Python
from sympy import isprime, primefactors for k in range(10**6): s0, s1 = sum(primefactors(k)), sum(primefactors(k + 1)) if s0 == s1 and isprime(s0): print(k, end=', ') # Jason Yuen, Jun 05 2024
Extensions
More terms from Michel Marcus, May 11 2024
Comments