A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.

2, 3, 4, 9, 20, 24, 98, 170, 1104, 1274, 2079, 2255, 3438, 4233, 4345, 4716, 5368, 7105, 7625, 10620, 13350, 13775, 14905, 20220, 21385, 23408, 25592, 26123, 28518, 30457, 34945, 35167, 38180, 45548, 49230, 51911, 52206, 53456, 56563, 61456, 65429, 66585

Offset: 1

Author

Abhiram R Devesh, Aug 11 2013

Keywords

• easy
• nonn

Comments

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

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,...

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

The sum of reciprocals of this sequence is approximately equal to 1.3077. - Abhiram R Devesh, Jun 14 2014

Examples

			For n=20: prime factors = 2,2,5; sum of prime factors = 9.
For n+1=21: prime factors = 3,7; sum of prime factors = 10.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..300
• Giovanni Resta, eRAPs: the 446139 terms < 10^12
• Carlos Rivera, Extension to Ruth Aaron pairs

Crossrefs

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

Programs

Mathematica

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 *)

PARI

sopfm(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1]*f[i,2])
for(n=1,10^5,if(sopfm(n)==sopfm(n+1)-1,print1(n,","))) /* Ralf Stephan, Aug 12 2013 */

Python

## sumdivisors(n) is a function that would return the sum of prime
## divisors of n. (See A001414)
i=2
while i < 100000:
  sdi=sumdivisors(i)
  sdip=sumdivisors(i+1)
  if sdi==sdip-1:
    print(i)
  i=i+1

Extensions

More terms from Ralf Stephan, Aug 12 2013