OFFSET
1,4
COMMENTS
Sum of the prime numbers among the smallest parts of the partitions of n into two parts. For example, a(8)=5; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). The prime numbers among the smallest parts are 2 and 3, so 2 + 3 = 5. - Wesley Ivan Hurt, Nov 01 2017
Number of distinct rectangles with integer length and prime width such that L + W = n, W <= L. For a(14)=17; the rectangles are 2 X 12, 3 X 11, 5 X 9, and 7 X 7. The sum of the lengths are then 2+3+5+7 = 17. - Wesley Ivan Hurt, Nov 08 2017
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = Sum_{i=1..floor(n/2)} i * A010051(i). - Wesley Ivan Hurt, Oct 31 2017
a(n) = A034387(floor(n/2)) for n >= 2. - Georg Fischer, Nov 28 2022
a(n) = A063958(n!). - Amiram Eldar, Jul 24 2024
EXAMPLE
20! = (2^18)*(3^8)*(5^4)*(7^2)*11*13*17*19, the non-unitary prime divisors are {2, 3, 5, 7}, so a(20) = 2 + 3 + 5 + 7 = 17.
MAPLE
seq(add(j, j=select(isprime, [$1..iquo(n, 2)])), n=1..65); # Peter Luschny, Nov 28 2022
MATHEMATICA
Join[{0, 0, 0}, Table[Total[Transpose[Select[FactorInteger[n!], Last[#]>1&]][[1]]], {n, 4, 70}]] (* Harvey P. Dale, Jun 19 2013 *)
PROG
(PARI) { for (n=1, 1000, f=factor(n!)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063960.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Sep 04 2001
STATUS
approved