A295663 - OEIS

%I #19 Sep 28 2023 04:26:27

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

%H Antti Karttunen, <a href="/A295663/b295663.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

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

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

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

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

%o (Scheme, with memoization-macro definec)

%o (definec (A295663 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 0 (A007814 (+ 1 (A067029 n)))) (A295663 (A028234 n)))))

%o (define (A295663 n) (- (A295664 n) (A056169 n)))

%o (PARI) a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ _Amiram Eldar_, Sep 28 2023

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

%Y Cf. A295661 (positions of nonzero terms).

%K nonn,easy

%O 1,8

%A _Antti Karttunen_, Nov 28 2017