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

%I A295663 #19 Sep 28 2023 04:26:27
%S A295663 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,
%T A295663 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,
%U A295663 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
%N A295663 a(n) = A295664(n) - A056169(n); 2-adic valuation of tau(n) minus the number of unitary prime divisors of n.
%H A295663 Antti Karttunen, <a href="/A295663/b295663.txt">Table of n, a(n) for n = 1..65537</a>
%H A295663 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A295663 Additive with a(p) = 0, a(p^e) = A007814(1+e) if e > 1.
%F A295663 a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A007814(1+A067029(n)) + a(A028234(n)).
%F A295663 a(n) = A295664(n) - A056169(n).
%F A295663 a(n) = 0 iff A295662(n) = 0, and when A295662(n) > 0, a(n) >= A295662(n).
%F A295663 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
%t A295663 Table[IntegerExponent[DivisorSigma[0, n], 2] - DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* _Michael De Vlieger_, Nov 28 2017 *)
%o A295663 (Scheme, with memoization-macro definec)
%o A295663 (definec (A295663 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 0 (A007814 (+ 1 (A067029 n)))) (A295663 (A028234 n)))))
%o A295663 (define (A295663 n) (- (A295664 n) (A056169 n)))
%o A295663 (PARI) a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ _Amiram Eldar_, Sep 28 2023
%Y A295663 Cf. A000005, A007814, A056169, A295662, A295664.
%Y A295663 Cf. A295661 (positions of nonzero terms).
%K A295663 nonn,easy
%O A295663 1,8
%A A295663 _Antti Karttunen_, Nov 28 2017