A228126 - OEIS

%I #39 Sep 07 2016 10:27:32

%S 2,3,4,9,20,24,98,170,1104,1274,2079,2255,3438,4233,4345,4716,5368,

%T 7105,7625,10620,13350,13775,14905,20220,21385,23408,25592,26123,

%U 28518,30457,34945,35167,38180,45548,49230,51911,52206,53456,56563,61456,65429,66585

%N Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.

%C This is an extension to Ruth-Aaron pairs. Sum of prime factors (inclusive of multiplicity) of pair of Consecutive positive integers are also consecutive.

%C The number of pairs less than 10^k (k=1,2,3,4,5,6,..) with this property are 4,7,8,19,55,149,...

%C Up to 10^13 there are only 5 sets of consecutive terms, namely, (2, 3), (3,4), (27574665988, 27574665989), (862179264458, 1862179264459) and (9600314395008, 9600314395009). - _Giovanni Resta_, Dec 24 2013

%C The sum of reciprocals of this sequence is approximately equal to 1.3077. - _Abhiram R Devesh_, Jun 14 2014

%H Harvey P. Dale, <a href="/A228126/b228126.txt">Table of n, a(n) for n = 1..300</a>

%H Giovanni Resta, <a href="http://www.numbersaplenty.com/set/eRAP/">eRAPs: the 446139 terms < 10^12</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_719.htm">Extension to Ruth Aaron pairs</a>

%e For n=20: prime factors = 2,2,5; sum of prime factors = 9.

%e For n+1=21: prime factors = 3,7; sum of prime factors = 10.

%t spd[n_]:=Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]]; Rest[ Position[ Partition[Array[spd,70000],2,1],_?(#[[2]]-#[[1]]==1&),{1}, Heads->False]//Flatten] (* _Harvey P. Dale_, Sep 07 2016 *)

%o (Python)

%o ## sumdivisors(n) is a function that would return the sum of prime

%o ## divisors of n.

%o i=2

%o while i < 100000:

%o ..sdi=sumdivisors(i)

%o ..sdip=sumdivisors(i+1)

%o ..if sdi==sdip-1:

%o ....print i,i+1

%o ..i=i+1

%o (PARI) sopfm(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1]*f[i,2])

%o for(n=1,10^5,if(sopfm(n)==sopfm(n+1)-1,print1(n,","))) /* _Ralf Stephan_, Aug 12 2013 */

%Y Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.

%K easy,nonn

%O 1,1

%A _Abhiram R Devesh_, Aug 11 2013

%E More terms from _Ralf Stephan_, Aug 12 2013