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

1,8

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n.

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

Mathematica

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

Prog

(Scheme, with memoization-macro definec)
(definec (A295663 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 0 (A007814 (+ 1 (A067029 n)))) (A295663 (A028234 n)))))
(define (A295663 n) (- (A295664 n) (A056169 n)))
(PARI) a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Crossrefs

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

Cf. A295661 (positions of nonzero terms).

Sequence in context: A014083 A238403 A112315 * A005926 A353419 A346483

Adjacent sequences: A295660 A295661 A295662 * A295664 A295665 A295666

Keyword

nonn,easy

Author

Antti Karttunen, Nov 28 2017

Status

approved