A364988 -id:A364988 - cp's OEIS frontend

A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

30, 42, 66, 126, 70, 78, 198, 264, 90, 234, 126, 120, 462, 312, 270, 270, 306, 150, 154, 696, 798, 936, 390, 210, 210, 290, 210, 714, 210, 286, 210, 462, 744, 240, 1710, 1224, 910, 330, 420, 2262, 390, 270, 714, 870, 1050, 294, 330, 630, 630, 2232, 378, 1620, 330

Offset: 1

Michael De Vlieger, Jan 13 2025

nonn

Define function f(x) to be 1/2 * card({ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) }), a counting function of distinct divisor pairs (d, x/d) that are both coreful but neither divides the other.
Define function g(x) to be Sum_{ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) } d.
Define h(x) to be equal to A364988(x) = Sum_{ d | x : rad(d) = rad(x) } d, sum of coreful divisors of x.
a(n) = g(A376936(n)) <= h(A376936(n)).
The function f(x) is analogous to tau(x) = A000005(x) while g(x) is analogous to sigma(x) = A000203(x).
f(k) > 0 and g(k) > 1 for k in A376936, otherwise f(k) = 0 and g(k) = 0.
Since rad(d) = rad(k/d) = rad(k), a(n) = m*rad(k), with integer m > 1.

			Let s = A376936.
a(1) = 30 since s(1) = 216 = 12*18 = 2*6 + 3*6 = 5*rad(216), and the sum of these is 30.
a(2) = 42 since s(2) = 432 = 18*24 = 3*6 + 4*6 = 7*rad(432), and the sum of these is 42.
a(3) = 66 since s(3) = 648 = 12*54 = 2*6 + 9*6 = 11*rad(648), and their sum is 66.
a(4) = 126 since s(4) = 864 = 18*48 = 24*36, and the sum of all these divisors is 126, etc. Note that 18 + 48 + 24 + 36 = 3*6 + 8*6 + 4*6 + 6*6 = 21*rad(864).

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

Cf. A364988, A379552.

nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[! MemberQ[{1, #1, #2}, GCD @@ {##}],
      rad[#1] == rad[#2]] & @@ {#, n/#} &]], s]

A379523 Sum of coreful divisors d | k such that gcd(d, k/d) > 1 and rad(d) = rad(k/d), with d | k/d and d < d/k, where k is in A320966 and rad = A007947.

Original entry on oeis.org

6, 10, 12, 30, 54, 18, 30, 24, 30, 126, 30, 30, 42, 120, 238, 90, 60, 56, 42, 50, 126, 60, 510, 162, 130, 168, 60, 336, 70, 150, 234, 66, 240, 110, 990, 90, 378, 432, 84, 132, 78, 112, 210, 270, 546, 90, 110, 456, 330, 150, 2046, 1092, 182, 714, 102, 350, 260

Offset: 1

Michael De Vlieger, Jan 15 2025

nonn

Alternatively, sum of divisors d | k such that omega(d) = omega(k/d) = omega(k), d | k/d, and d < k/d, where omega = A001221.

			a(1) = 6 since s(1) = 8 = 2*4; 2 | 4 but 4 > 2; 2+4 = 6. a(1) = (2^3-1)/(2-1)-1 = 6.
a(2) = 10 since s(2) = 16 = 2*8; 2 | 8 but 8 > 2; 2+8 = 10. a(2) = (2^4-1)/(2-1)-(2^2)-1 = 10.
a(3) = 12 since s(3) = 27 = 3*9; 3 | 9 but 9 > 3; 3+9 = 12. a(3) = (3^3-1)/(3-1)-1 = 12.
a(4) = 30 since s(4) = 32 = (2^5-1)/(2-1)-1 = 30.
a(5) = 54 since s(5) = 64 = (2^6-1)/(2-1)-(2^3)-1 = 54.
a(6) = 18 since s(6) = 72 = 6*12; 6 | 12 but 12 > 6; 6+12 = 18.
a(8) = 24 since s(8) = 108 = 6*18; 6 | 24 but 24 > 6; 6+18 = 24, etc.

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.

Cf. A000079, A001221, A007947, A246549, A320966, A364988.

nn = 2500;
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] > 0 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[PrimeNu[#1] == PrimeNu[#2] == #3,
        Xor[Divisible[#2, #1], Divisible[#1, #2]]] & @@
    {#, n/#, PrimeNu[n]} &]], s]

Let s(n) = A320966(n).
a(n) <= A364988(s(n)).
Using Iverson brackets:
For s(n) = 2^e, e > 2, (i.e., in A000079):
a(n) = 2^e - [e mod 2 = 0]*sqrt(2^e)-2
     = 2^e - 2*(e-1) for even e or 2^e-2 for odd e.
     = A364988(s(n)) - [e mod 2 = 0]*2^(e/2) for s(n) = 2^e, e > 2.
For s(n) = p^e, e > 2, (i.e., in A246549):
a(n) = A364988(s(n)) - [e mod 2 = 0]*p^(e/2)
     = (p^e - 1)/(p-1) - [e mod 2 = 0]*sqrt(p^e) - 1.

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

Original entry on oeis.org

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

Michael De Vlieger, Jan 15 2025

nonn

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.

			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.

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.

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

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]

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.

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A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

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A379523 Sum of coreful divisors d | k such that gcd(d, k/d) > 1 and rad(d) = rad(k/d), with d | k/d and d < d/k, where k is in A320966 and rad = A007947.

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

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