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Integer log of n, sum of prime factors of n (with multiplicity)
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n 
sopfr (n) 
n 

sopfr (n) := ω (n)∑ i = 1ω (n)∑ i = 1
where

n = ω (n)∏ i = 1
Ω (n) 
n 
n 
n ≥ 1 
 {0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, ...}
Contents
Iterated integer log of n
A029908 Starting withn 
 {0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, 5, 5, 17, 5, 19, 5, 7, 13, 23, 5, 7, 5, 5, 11, 29, 7, 31, 7, 5, 19, 7, 7, 37, 7, 5, 11, 41, 7, 43, 5, 11, 7, 47, 11, 5, 7, 5, 17, 53, 11, 5, 13, 13, 31, 59, 7, 61, 5, 13, 7, 5, 5, 67, 7, 5, 5, 71, 7, 73, 5, 13, 23, 5, 5, 79, 13, 7, 43, 83, 5, 13, ...}
%C That is, the sopfr function (see A001414) applied repeatedly until reaching 0 or a fixed point. %C For n > 1 the sequence reaches a fixed point which is either 4 or a prime. %C A002217(n) is number of terms in sequence from n to a(n).  Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 08 2003 %C Because sopfr(n) <= n (with equality at 4 and the primes), the first appearance of all primes is in the natural order: 2,3,5,7,11,... . [Zak Seidov, Mar 14 2011]
Primes p such that sopf ( p − 1) = sopf ( p + 1)
A086711 Primesp 
(p − 1) 
(p + 1) 
n 
 {11, 17, 31, 251, 1429, 3041, 16561, 16927, 53299, 56897, 89783, 95089, 213599, 282977, 345547, 432587, 592223, 763457, 906949, 915799, 1050449, 1058389, 1485017, 1577341, 1678399, 1780253, ...}
Conjecture: sequence is infinite.
 Forwarded message  From: "N. J. A. Sloane" <njas@research.att.com> To: seqfan@list.seqfan.eu Date: Mon, 16 May 2011 17:38:27 0400 Subject: [seqfan] Re: sum of prime factors of p1 and p+1 But A086711 doesn't require that the sum of prime factors of p1 and p+1 be a prime  although it IS in all the terms shown! Is it always? If so, why? If not, there should be another sequence (or two) with the other version and the exceptions. Neil
Sequences
A126975 Primesp 
q 
p + q 
 {2, 5, 23, 43, 83, 97, 103, 131, 149, 157, 179, 191, 193, 229, 251, 293, 337, 383, 397, 401, 431, 443, 463, 541, 569, 601, 643, 709, 739, 857, 859, 863, 887, 907, 911, 967, 971, 983, 1019, 1039, 1069, ...}
p 
sopfr ( p − 1) = sopfr ( p + 1) 
sopfr (n) 
 {11, 251, 1429, 906949, 1050449, 1058389, 3728113, 9665329, 13623667, 14320489, 30668003, 30910391, 45717377, 49437001, 55544959, 57510911, 58206653, 58772257, 69490901, 72191321, ...}
See also
 {{Sum of prime factors (with multiplicity)}} arithmetic function template
 Integer log of n, sum of prime factors of n (with multiplicity)
 Integer log of n!, sum of prime factors of n! (with multiplicity)
 Prime factors of n or distinct prime factors of n
 Number of prime factors of n or number of distinct prime factors of n (ω (n))
 Sum of prime factors of n or sum of distinct prime factors of n (sopf (n) or sodpf (n))
 Product of prime factors of n or product of distinct prime factors of n (radical of n, rad (n)) (squarefree kernel of n)
 Prime factors of n (with multiplicity)
 Number of prime factors of n (with multiplicity) (Ω (n))
 Sum of prime factors of n (with repetition) (sopfr (n)) (integer log of n)
 Product of prime factors of n (with repetition) (n, positive integers)
External links
 Weisstein, Eric W., Sum of Prime Factors, from MathWorld—A Wolfram Web Resource.