A328957 Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 42, 43, 47, 49, 53, 59, 61, 64, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 89, 97, 100, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 120, 121, 125, 127, 128, 130, 131, 137

Offset: 1

Author

Gus Wiseman, Nov 01 2019

Keywords

• nonn

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}
		

Links

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

Crossrefs

Nonzeros of A328958.

The complement is A328956.

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. A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Programs

Mathematica

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

PARI

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

Formula

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