A328958 a(n) = d(n) - (omega(n) * bigomega(n)), where d (number of divisors) = A000005, omega = A001221, bigomega = A001222.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 2, 1, 0, 0, 0, 0, -1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, -1, 1, 0, -1

Offset: 1

Author

Gus Wiseman, Nov 02 2019

Keywords

• sign
• easy

Comments

a(n) = sigma_0(n) - omega(n) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222. - The original name of the sequence.

Examples

			a(144) = sigma_0(144) - omega(144) * nu(144) = 15 - 6 * 2 = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Positions of first appearances are A328962.

Zeros are A328956.

Nonzeros are A328957.

omega(n) * nu(n) is A113901(n).

(omega(n) - 1) * nu(n) is A307409(n).

sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).

Cf. A000005, A001221, A001222, A112798, A124010, A323023, A328960, A328961, A328963, A328964.

Programs

Mathematica

Table[DivisorSigma[0,n]-PrimeOmega[n]*PrimeNu[n],{n,100}]

PARI

A328958(n) = (numdiv(n)-(omega(n)*bigomega(n))); \\ Antti Karttunen, Jan 27 2025

Formula

a(n) = A000005(n) - A001222(n) * A001221(n) = A000005(n) - A113901(n).

Extensions

More terms added and the function names in the definition replaced with standard OEIS ones - Antti Karttunen, Jan 27 2025