|
|
A067392
|
|
Sum of numbers <= n which have common prime factors with n.
|
|
11
|
|
|
0, 2, 3, 6, 5, 15, 7, 20, 18, 35, 11, 54, 13, 63, 60, 72, 17, 117, 19, 130, 105, 143, 23, 204, 75, 195, 135, 238, 29, 345, 31, 272, 231, 323, 210, 450, 37, 399, 312, 500, 41, 651, 43, 550, 495, 575, 47, 792, 196, 775, 510, 754, 53, 999, 440, 924, 627, 899, 59
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Sum of k <= n such that gcd(n,k) > 1.
|
|
LINKS
|
|
|
FORMULA
|
Not multiplicative.
a(p) = p where p is a prime; a(2^k) = 2^(k-1)*{2^(k-1) + 1).
Sum_{k=1..n} a(k) ~ (1/6 - 1/(Pi^2)) * n^3. - Amiram Eldar, Dec 03 2023
|
|
EXAMPLE
|
For n=24, a(24) = 2+3+4+6+8+9+10+12+14+15+16+18+20+21+22+24 = 204.
|
|
MATHEMATICA
|
a[n_] := Plus@@Select[Range[1, n], GCD[ #, n]>1&]
Join[{0}, Table[n (n + 1) / 2 - n EulerPhi@(n) / 2, {n, 2, 60}]] (* Vincenzo Librandi, Jul 19 2019 *)
|
|
PROG
|
(PARI) A067392(n)={a=0; for(i=1, n, if(gcd(i, n)<>1, a=a+i)); a}
(PARI) a(n) = sum(k=1, n, k*(gcd(k, n) != 1)); \\ Michel Marcus, May 08 2018
(PARI) a(n) = if(n == 1, 0, n*(n + 1 - eulerphi(n))/2); \\ Amiram Eldar, Dec 03 2023
(Magma) [0] cat [n*(n+1)/2-n*EulerPhi(n)/2: n in [2..60]]; // Vincenzo Librandi, Jul 19 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|