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# Sum of distinct prime factors of n

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The sum of distinct prime factors of n sopf(n) is given by

${\displaystyle {\rm {sopf}}(n)\equiv \sum _{i=1}^{\omega (n)}p_{i}}$

where

${\displaystyle n=\prod _{i=1}^{\omega (n)}{p_{i}}^{\alpha _{i}}}$

and ${\displaystyle \scriptstyle \omega (n)}$ is the number of distinct prime factors of n.

 257     ${\displaystyle \scriptstyle {\text{sopf}}(n)\,}$
0

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

A008472 Sum of distinct primes dividing ${\displaystyle \scriptstyle n\,}$.

{0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 8, 2, 17, 5, 19, 7, 10, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 14, 19, 12, 5, 37, 21, 16, 7, 41, 12, 43, 13, 8, 25, 47, 5, 7, 7, 20, 15, 53, 5, 16, 9, 22, 31, 59, 10, 61, 33, ...}

## Sequences

A190722 Primes ${\displaystyle \scriptstyle p\,}$ such that A008472${\displaystyle \scriptstyle (p-1)\,}$ = A008472${\displaystyle \scriptstyle (p+1)\,}$ and is a prime.

{3, 45751, 149351, 171529, 223099, 434237, 678077, 706841, 1996297, 3993037, 6340457, 7199113, 7419761, 9000317, 13129271, 15052777, 17193217, 18436879, 18749881, 18998519, ...}