A295663 a(n) = A295664(n) - A056169(n); 2-adic valuation of tau(n) minus the number of unitary prime divisors of n.

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

Offset: 1

Author

Antti Karttunen, Nov 28 2017

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A007814, A056169, A295662, A295664.

Cf. A295661 (positions of nonzero terms).

Programs

Mathematica

Table[IntegerExponent[DivisorSigma[0, n], 2] - DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)

PARI

a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Formula

Additive with a(p) = 0, a(p^e) = A007814(1+e) if e > 1.

a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A007814(1+A067029(n)) + a(A028234(n)).

a(n) = A295664(n) - A056169(n).

a(n) = 0 iff A295662(n) = 0, and when A295662(n) > 0, a(n) >= A295662(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.22852676306472099280..., where f(x) = -1 + (1-x)*(-x + Sum_{k>=0} x^(2^k-1)/(1-x^(2^k))). - Amiram Eldar, Sep 28 2023