

A228126


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


8



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
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OFFSET

1,1


COMMENTS

This is an extension to RuthAaron 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


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


EXAMPLE

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.


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 *)


PROG

(Python)
## sumdivisors(n) is a function that would return the sum of prime
## divisors of n.
i=2
while i < 100000:
..sdi=sumdivisors(i)
..sdip=sumdivisors(i+1)
..if sdi==sdip1:
....print i, i+1
..i=i+1
(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 */


CROSSREFS

Cf. A001414, A006145 RuthAaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
Sequence in context: A243902 A086865 A258274 * A352197 A192988 A280016
Adjacent sequences: A228123 A228124 A228125 * A228127 A228128 A228129


KEYWORD

easy,nonn


AUTHOR

Abhiram R Devesh, Aug 11 2013


EXTENSIONS

More terms from Ralf Stephan, Aug 12 2013


STATUS

approved



