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

1,1

Comments

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).

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

Links

Table of n, a(n) for n=1..37.

Numberphile, Aaron Numbers.

Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.

Wikipedia, Ruth-Aaron pair.

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Subsequence of A006145.

Cf. A008472.

Sequence in context: A063143 A006145 A219509 * A202326 A085646 A121546

Adjacent sequences: A372452 A372453 A372454 * A372456 A372457 A372458

Keyword

nonn,easy

Author

Gerardo Salcido Martinez, May 01 2024

Extensions

More terms from Michel Marcus, May 11 2024

Status

approved