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Degree of divisor lattice D(n).
5

%I #48 Feb 13 2024 02:19:55

%S 0,1,1,2,1,2,1,2,2,2,1,3,1,2,2,2,1,3,1,3,2,2,1,3,2,2,2,3,1,3,1,2,2,2,

%T 2,4,1,2,2,3,1,3,1,3,3,2,1,3,2,3,2,3,1,3,2,3,2,2,1,4,1,2,3,2,2,3,1,3,

%U 2,3,1,4,1,2,3,3,2,3,1,3,2,2,1,4,2,2,2,3,1,4,2,3,2,2,2,3,1,3,3,4,1,3,1,3,3

%N Degree of divisor lattice D(n).

%H Antti Karttunen, <a href="/A238949/b238949.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Sung-Hyuk Cha)

%H Sung-Hyuk Cha, Edgar G. DuCasse, and Louis V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arXiv:1405.5283 [math.NT], 2014.

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

%F a(n) = A001221(n) + A056170(n) as given in the Cha, DuCasse, Quintas reference. - _Geoffrey Critzer_, Mar 02 2015

%F Additive with a(p^e) = 1+A057427(e-1). - _Antti Karttunen_, Jul 23 2017

%F Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^2 (A085548). - _Amiram Eldar_, Feb 13 2024

%t Prepend[Table[Total[FactorInteger[n][[All, 2]] /. x_ /; x > 1 -> 2], {n, 2, 85}], 0] (* _Geoffrey Critzer_, Mar 02 2015 *)

%o (PARI) a(n) = {my(f = factor(n)); sum(i=1, #f~, 1 + (f[i,2] > 1));} \\ _Michel Marcus_, Mar 03 2015

%o (Scheme) (define (A238949 n) (if (= 1 n) 0 (+ 1 (A057427 (+ -1 (A067029 n))) (A238949 (A028234 n))))) ;; _Antti Karttunen_, Jul 23 2017

%Y Cf. A001221, A056170, A057427, A077761, A085548.

%K nonn,easy

%O 1,4

%A _Sung-Hyuk Cha_, Mar 07 2014

%E More terms from _Antti Karttunen_, Jul 23 2017