A063955 Sum of the unitary prime divisors of n!.

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, 59, 59, 59, 88, 88, 119, 119, 119, 102, 102, 102, 139, 120, 120, 120, 161, 161, 204, 204, 204, 181, 228, 228, 228, 228, 228, 228, 281, 281, 281, 281, 281, 252, 311, 311

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Formula

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

a(n) = A063956(n!). - Amiram Eldar, Jul 24 2024

Example

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).)

Maple

a:= n-> add(`if`(i[2]=1, i[1], 0), i=ifactors(n!)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 24 2018

Mathematica

a[n_] := Select[FactorInteger[n!], #[[2]] == 1&][[All, 1]] // Total;
Array[a, 60] (* Jean-François Alcover, Jan 01 2022 *)

Prog

(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

Crossrefs

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063956.

Sequence in context: A338060 A169856 A185112 * A097594 A204980 A300896

Adjacent sequences: A063952 A063953 A063954 * A063956 A063957 A063958

Keyword

nonn,easy

Author

Labos Elemer, Sep 04 2001

Status

approved