A063960 - OEIS

%I #41 Jul 24 2024 21:30:36

%S 0,0,0,2,2,5,5,5,5,10,10,10,10,17,17,17,17,17,17,17,17,28,28,28,28,41,

%T 41,41,41,41,41,41,41,58,58,58,58,77,77,77,77,77,77,77,77,100,100,100,

%U 100,100,100,100,100,100,100,100,100,129,129,129,129,160,160,160,160

%N Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

%C 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

%C 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

%H Harry J. Smith, <a href="/A063960/b063960.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{i=1..floor(n/2)} i * A010051(i). - _Wesley Ivan Hurt_, Oct 31 2017

%F a(n) = A034387(floor(n/2)) for n >= 2. - _Georg Fischer_, Nov 28 2022

%F a(n) = A063958(n!). - _Amiram Eldar_, Jul 24 2024

%e 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.

%p seq(add(j, j=select(isprime, [$1..iquo(n,2)])), n=1..65); # _Peter Luschny_, Nov 28 2022

%t Join[{0,0,0},Table[Total[Transpose[Select[FactorInteger[n!], Last[#]>1&]][[1]]],{n,4,70}]] (* _Harvey P. Dale_, Jun 19 2013 *)

%o (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

%Y Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063958.

%K nonn,easy

%O 1,4

%A _Labos Elemer_, Sep 04 2001