A084227 -id:A084227 - cp's OEIS frontend

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

First differs from A084227 in having 60.

			The sequence of terms together with their prime indices begins:
   6: {1,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  39: {2,6}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Zeros of A328958.
The complement is A328957.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - omega(n) * Omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A000040, A005117, A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]==PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) == omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) = A001222(a(n)) * A001221(a(n)).

A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 2, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 7, 1, 2, 2, 0, 2, 5, 1, 2, 2, 5, 1, 4, 1, 2, 2, 2, 2, 5, 1, 2, 0, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(72) = 4 factorizations are (2*3*12), (3*24), (6*12), (72). Missing from this list and not strict are (2*2*2*3*3), (2*2*3*6), (2*6*6), (2*2*18), while missing from the list and using perfect powers are (2*36), (2*4*9), (3*4*6), (4*18), (8*9).

Positions of 0's are A246547.
Positions of 1's are A000040.
Positions of 2's are A084227.
Positions of 3's are A085986.
Positions of 4's are A143610.
Cf. A001055, A001597, A007916, A025147, A045778, A052410, A303707, A323054, A323087, A323088, A323089.

radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
facssr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facssr[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[facssr[n]],{n,100}]

A082878 Arithmetic derivative of numbers of the form p*q^k with distinct primes p and q, k > 0.

Original entry on oeis.org

5, 7, 16, 9, 8, 21, 24, 10, 13, 44, 15, 32, 14, 19, 12, 21, 16, 68, 48, 39, 25, 112, 45, 20, 56, 81, 16, 92, 22, 31, 33, 51, 18, 72, 26, 39, 55, 80, 18, 176, 43, 22, 45, 32, 140, 20, 96, 34, 49, 24, 272, 77, 75, 164, 55, 40, 240, 28, 120, 87, 61, 24, 63, 44, 128, 46, 26, 69

Offset: 1

Reinhard Zumkeller, May 25 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003415, A084227.

s[n_] := Module[{f = FactorInteger[n], i}, If[Length[f] == 2 && Min[f[[;;, 2]]] == 1, i = FirstPosition[f[[;;, 2]], 1][[1]]; n * (1/f[[i, 1]] + f[[3-i, 2]] /f[[3-i, 1]]), Nothing]]; Array[s, 100] (* Amiram Eldar, Mar 25 2025 *)

a(n) = A003415(A084227(n)).

a(p*q^k) = (p+q)*q^(k-1).

Data corrected by Amiram Eldar, Mar 25 2025

A105642 Composite nonsquares and noncubes.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 106, 108

Offset: 1

Zak Seidov, May 03 2005

nonn

Cf. A007774, A024619, A030231, A056760, A064040, A084227, A100658.

nn=200; Complement[Range[nn], Prime[Range[PrimePi[nn]]], Range[Sqrt[nn]]^2, Range[nn^(1/3)]^3]  (* Harvey P. Dale, Jan 26 2011 *)

is(n)=n>1 && !isprime(n) && !issquare(n) && !ispower(n,3) \\ Charles R Greathouse IV, Oct 19 2015

a(n) = n + O(n/log n). - Charles R Greathouse IV, Oct 19 2015

cp's OEIS Frontend

A328956 Numbers k such that sigma_0(k) = omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A323090 Number of strict factorizations of n using elements of A007916 (numbers that are not perfect powers).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A082878 Arithmetic derivative of numbers of the form p*q^k with distinct primes p and q, k > 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A105642 Composite nonsquares and noncubes.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula