A380144 Sum of divisors d | k such that rad(d) = rad(k/d) where k is in A001694 and rad = A007947.

1, 2, 6, 3, 14, 5, 12, 30, 6, 7, 62, 18, 39, 10, 24, 11, 30, 126, 42, 13, 14, 30, 72, 15, 120, 254, 90, 17, 78, 56, 19, 42, 70, 168, 21, 22, 60, 510, 23, 186, 155, 234, 60, 26, 363, 98, 150, 29, 360, 30, 31, 66, 240, 180, 1022, 33, 90, 378, 34, 35, 546, 84, 132

Offset

1,2

Comments

Alternatively, sum of divisors d | k such that omega(d) = omega(k/d) = omega(k) for powerful k in A001694.

Not a permutation of natural numbers: 4 does not appear.

This sequence is A364988 without zeros. See A364988 for multiplicative properties.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5.

Formula

Let s = A001694.

a(n) = A364988(A001694(n)).

a(n) = m*rad(n) with m >= 1.

a(n) = sqrt(s(n)) = rad(s(n)) for s(n) = rad(s(n))^2 (i.e., s(n) in A062503).

a(n) > sqrt(s(n)) and a(n) = m*rad(s(n)), m > 1, for s(n) in A320966.

The sequence of record setters is A151821.

a(n) = s(n)-2 for s(n) in A151821.

Example

a(1) = 1 since s(1) = 1 = 1*1. (We count divisor 1 only once, since it is a square root.)
a(2) = 2 since s(2) = 4 = 2*2, and omega(2) = omega(4). Alternatively, rad(2) = rad(4).
a(3) = 6 since s(3) = 8 = 2*4, omega(2) = omega(4) = omega(8), i.e., rad(2) = rad(4) = rad(8), and 2+4 = 6.
a(4) = 3 since s(4) = 9 = 3*3, and omega(3) = omega(9); rad(3) = rad(9).
a(5) = 14 since s(5) = 16 = 2*8 = 4*4; 2+4+8 = 14.
a(6) = 5 since s(6) = 25 = 5*5.
a(7) = 12 since s(7) = 27 = 3*9; 3+9 = 12.
a(8) = 30 since s(8) = 32 = 2*16 = 4*8; 2+4+8+16 = 30.
a(9) = 6 since s(9) = 36 = 6*6, etc.

Mathematica

nn = 1500;
s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];
Map[Function[{n, r},
  DivisorSum[n, # &, PrimeNu[#] == PrimeNu[n/#] == r &]] @@
  {#, PrimeNu[#]} &, s]

Crossrefs

Cf. A001221, A001694, A007947, A062503, A151821, A320966, A364988.

Sequence in context: A120859 A253258 A098810 * A360006 A081469 A331441

Adjacent sequences: A380141 A380142 A380143 * A380145 A380146 A380147

Keyword

nonn

Author

Michael De Vlieger, Jan 15 2025

Status

approved