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A228126
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Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.
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4
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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
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1,1
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COMMENTS
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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
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LINKS
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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==sdip-1:
....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 */
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CROSSREFS
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Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
Sequence in context: A243902 A086865 A258274 * A192988 A280016 A122534
Adjacent sequences: A228123 A228124 A228125 * A228127 A228128 A228129
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KEYWORD
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easy,nonn
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AUTHOR
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Abhiram R Devesh, Aug 11 2013
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EXTENSIONS
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More terms from Ralf Stephan, Aug 12 2013
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STATUS
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approved
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