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

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The sum of prime factors of

n

(with repetition) (

sopfr (n)

), or integer log of

n

, is given by

sopfr (n)  := 
 
ω (n)
∑
 
i  = 1
  
(α i  ) 1 ⋅  pi  = 
 
ω (n)
∑
 
i  = 1
  
α i ⋅  pi

where

n  = 
 
ω (n)
∏
 
i  = 1
  
pi α i

and

Ω (n)

is the number of distinct prime factors of n.


A001414 Integer log of

n

: sum of primes factors (sopfr) of

n

(with repetition),

n   ≥   1

. (In the graph of A001414, notice how the primes (and the twin primes) stand out, while the cousin primes ask for a bit more perusing.)

{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, ...}

Iterated integer log of n

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A029908 Starting with

n

, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point.

{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)

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A086711 Primes

p

such that A001414

 (p  −  1)

= A001414

 (p + 1)

, where A001414 is sum of primes dividing

n

(with repetition).

{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" <[email protected]>
To: [email protected]
Date: Mon, 16 May 2011 17:38:27 -0400
Subject: [seqfan] Re: sum of prime factors of p-1 and p+1
But A086711 doesn't require that the sum of prime factors of p-1 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

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A126975 Primes

p

with property that, if

q

is the next prime, then the sum of the prime factors of

p + q

, taken with multiplicity, is a prime.

{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, ...}

A190680 Primes

p

such that

sopfr (  p  −  1) = sopfr (  p  +  1)

is also prime, where

sopfr (n)

is A001414.

{11, 251, 1429, 906949, 1050449, 1058389, 3728113, 9665329, 13623667, 14320489, 30668003, 30910391, 45717377, 49437001, 55544959, 57510911, 58206653, 58772257, 69490901, 72191321, ...}

See also

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