A367202 If n = Product(p_i^e_i), a(n) = Sum_{i = 1..k}(rad(n)/p_i)^e_i, where rad is A007947.
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
Offset: 1
Keywords
Examples
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
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
- 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.
Programs
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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 *) -
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
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
Extensions
More terms from Michael De Vlieger, Nov 10 2023
Comments