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A063955 Sum of the unitary prime divisors of n!.

%I A063955 #27 Jul 24 2024 21:08:31
%S A063955 0,2,5,3,8,5,12,12,12,7,18,18,31,24,24,24,41,41,60,60,60,49,72,72,72,
%T A063955 59,59,59,88,88,119,119,119,102,102,102,139,120,120,120,161,161,204,
%U A063955 204,204,181,228,228,228,228,228,228,281,281,281,281,281,252,311,311
%N A063955 Sum of the unitary prime divisors of n!.
%H A063955 Alois P. Heinz, <a href="/A063955/b063955.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harry J. Smith)
%F A063955 a(n) = Sum_{k=floor(n/2)+1..n} k*c(k), where c is the prime characteristic (A010051). - _Wesley Ivan Hurt_, Dec 23 2023
%F A063955 a(n) = A063956(n!). - _Amiram Eldar_, Jul 24 2024
%e A063955 Prime divisors of 20! which have exponent 1 (i.e., unitary prime divisors) are {11, 13, 17, 19}, so a(20) = 11 + 13 + 17 + 19= 60. (The sum of all its prime divisors (unitary and non-unitary) is A034387(20).)
%p A063955 a:= n-> add(`if`(i[2]=1, i[1], 0), i=ifactors(n!)[2]):
%p A063955 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jun 24 2018
%t A063955 a[n_] := Select[FactorInteger[n!], #[[2]] == 1&][[All, 1]] // Total;
%t A063955 Array[a, 60] (* _Jean-François Alcover_, Jan 01 2022 *)
%o A063955 (PARI) a(n) = my(f=factor(n!)~); sum(i=1, length(f), if (f[2, i]==1, f[1, i])); \\ _Harry J. Smith_, Sep 04 2009
%Y A063955 Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063956.
%K A063955 nonn,easy
%O A063955 1,2
%A A063955 _Labos Elemer_, Sep 04 2001