|
| |
|
|
A367202
|
|
If n = Product(p_i^e_i), a(n) = Sum_{i = 1..k}(rad(n)/p_i)^e_i, where rad is A007947.
|
|
1
|
|
|
|
0, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 11, 1, 9, 8, 1, 1, 7, 1, 27, 10, 13, 1, 29, 1, 15, 1, 51, 1, 31, 1, 1, 14, 19, 12, 13, 1, 21, 16, 127, 1, 41, 1, 123, 28, 25, 1, 83, 1, 9, 20, 171, 1, 11, 16, 345, 22, 31, 1, 241, 1, 33, 52, 1, 18, 61, 1, 291, 26, 59, 1, 31, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,6
|
|
|
COMMENTS
|
|
|
|
LINKS
|
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, highlighting squareful numbers that are not prime powers in large light blue.
|
|
|
FORMULA
|
For n a prime power p^k, a(n) = (p/p)^1 = 1.
For n a squarefree semiprime a(n) = A001414(n).
For p,q distinct primes a(p*q^2) = q + p^2.
For n a squarefree number with prime divisors p_1,p_2..p_k, a(n) = Sum_{i = 1..k}(n/p_i) see Example
|
|
|
EXAMPLE
|
a(1) = 0, the empty sum.
rad(6) = rad(2*3) = 6 -->a(6) = (6/2)^1 + (6/3)^1 = 3 + 2 = 5.
rad(12) = rad(2^2*3) = 6 -->a(12) = (6/2)^2 + (6/3)^1 = 9 + 2 = 11.
rad(36) = rad(2^2*3^2) = 6 --> a(36) = (6/2)^2 +(6/3)^2 = 9 + 4 = 13.
rad(40) = rad(2^3*5^1) = 10 -->a(40) = (10/2)^3 + (10/5)^1 = 125 + 2 = 127.
n = 30 = 2*3*5 a squarefree number; a(30) = (30/2) + (30/3) + (30/5) = 15 + 10 + 6 = 31
|
|
|
MATHEMATICA
|
Array[Function[{r, w}, Total[Power @@@ Transpose@ {r/w[[All, 1]], w[[All, -1]]}]] @@ {Times @@ #[[All, 1]], #} &@ FactorInteger[#] &, 120] (* Michael De Vlieger, Nov 10 2023 *)
|
|
|
PROG
|
(PARI) rad(f) = factorback(f[, 1]);
a(n) = my(f=factor(n)); sum(i=1, #f~, (rad(f)/f[i, 1])^f[i, 2]); \\ Michel Marcus, Nov 10 2023
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
