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

1,6

Comments

Diverges from A028235 at a(12).

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.

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

Cf. A000040, A001414, A007947, A028235.

Sequence in context: A028235 A028236 A066504 * A292771 A369364 A322837

Adjacent sequences: A367199 A367200 A367201 * A367203 A367204 A367205

Keyword

nonn

Author

David James Sycamore, Nov 10 2023

Extensions

More terms from Michael De Vlieger, Nov 10 2023

Status

approved