A105221 - OEIS

%I #24 Sep 19 2023 08:21:06

%S 0,0,0,2,0,5,0,2,3,7,0,5,0,9,8,2,0,5,0,7,10,13,0,5,5,15,3,9,0,10,0,2,

%T 14,19,12,5,0,21,16,7,0,12,0,13,8,25,0,5,7,7,20,15,0,5,16,9,22,31,0,

%U 10,0,33,10,2,18,16,0,19,26,14,0,5,0,39,8,21,18,18,0,7,3,43,0,12,22,45

%N a(n) is the sum of n's distinct prime factors below n.

%H T. D. Noe, <a href="/A105221/b105221.txt">Table of n, a(n) for n=1..1000</a>

%F a(n) = A008472(n) - A010051(n) * n. - _Reinhard Zumkeller_, Apr 05 2013

%F G.f.: Sum_{k>=1} prime(k) * x^(2*prime(k)) / (1 - x^prime(k)). - _Ilya Gutkovskiy_, Apr 13 2021

%e a(12)=5 because 12's distinct prime factors 2 and 3 sum to 5.

%p f:= n -> convert(numtheory:-factorset(n) minus {n}, `+`):

%p map(f, [$1..100]); # _Robert Israel_, Sep 18 2023

%t Table[Total@Select[Join@@Union@*Table@@@FactorInteger@k,#<k&],{k,86}] (* _Giorgos Kalogeropoulos_, Nov 21 2021 *)

%o (Haskell)

%o a105221 n = a008472 n - n * fromIntegral (a010051 n)

%o -- _Reinhard Zumkeller_, Apr 05 2013

%o (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, if (f[k,1]<n, f[k,1])); \\ _Michel Marcus_, Nov 21 2021

%o (Python)

%o from sympy import primefactors

%o def A105221(n): return sum(p for p in primefactors(n) if p < n) # _Chai Wah Wu_, Sep 18 2023

%Y Cf. A003508, A008472, A010051.

%K easy,nonn

%O 1,4

%A _Alexandre Wajnberg_, Apr 13 2005

%E Edited by _Don Reble_, Nov 17 2005