A064142 - OEIS

%I #18 Jan 26 2023 21:31:26

%S 0,2,3,5,7,7,12,14,12,12,23,23,29,29,32,34,48,48,62,62,61,61,81,81,73,

%T 73,69,69,98,98,127,129,123,123,130,130,148,148,161,161,195,195,227,

%U 227,204,204,251,251,253,253,257,257,307,307,323,323,316,316,367,367

%N Sum of all distinct primes dividing central binomial coefficient C(n, floor(n/2)).

%H Harry J. Smith and Charles R Greathouse IV, <a href="/A064142/b064142.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Smith)

%F a(n) = A008472(A001405(n)).

%F k1 + o(1) < a(n)/(n^2/log n) < k2 + o(1) for k1 = 3/8 and k2 = 1/2. - _Charles R Greathouse IV_, Jan 26 2023

%t sop[n_] := If[n<2, 0, Total[First /@ FactorInteger[n]]]; Table[ sop[ Binomial[n, Floor[n/2]]], {n, 60}] (* _Giovanni Resta_, Jun 22 2018 *)

%o (PARI) sopf(n)= my(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s)

%o a(n)=sopf(binomial(n, n\2)); \\ _Harry J. Smith_, Sep 08 2009

%o (PARI) valp(n,p)=my(s); while(n\=p, s+=n); s

%o a(n)=my(s); forprime(p=2,n, my(t=valp(n,p)-valp(n\2,p)-valp(n-n\2,p)); if(t, s+=p)); s \\ _Charles R Greathouse IV_, Jan 26 2023

%Y Cf. A008472, A001405.

%K nonn

%O 1,2

%A _Labos Elemer_, Sep 11 2001