A371921 - OEIS

%I #9 Apr 12 2024 11:25:57

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

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

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

%N The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

%C Analogous to A098007 with A033880(n) = sigma(n) - 2*n instead of A001065(n) = sigma(n) - n.

%H Amiram Eldar, <a href="/A371921/b371921.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 if and only if n is nonabundant (A263837).

%F If a(n) > 0 then:

%F   a(n) > 1 if n is abundant (A005101).

%F   a(n) > 2 if n is in A371920.

%e a(n) = 0 if the iterations that start at n are entering a cycle. Examples of cycles are:

%e   1) Cycles of length 1: the triperfect numbers (A005820), 120, 672, 523776, ..., which are the fixed points of A033880. The triperfect numbers can be reached from other values of n, e.g., 276, 448, 486, 510, 702, ... .

%e   2) Cycles of length 2: the only known cycle is (45840, 51168) (see A069085). It can be reached from other values of n, e.g., 32130, 39420, 45480, 66300, ... .

%e   3) Cycles of length 3: the least cycle is (243732672, 271303776, 256786848). It is first reached from n = 107689320.

%e   4) Cycles of length 4: the least cycle is (65071776, 82842816, 89761152, 77260656). It can be reached from other values of n, e.g., 33623940, 41132280, 42825888, ... . The next cycle of length 4 is (985948800, 1381340160, 2183133696, 1489384608).

%t ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; a[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; Array[a, 120]

%o (PARI) ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k));}

%o a(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<<n); if(n == 0, hammingweight(t) - 1, 0);} \\ after _M. F. Hasler_ at A098007

%Y Cf. A000203 (sigma), A001065, A033880, A098007.

%Y Cf. A005101, A005820, A069085, A113285, A263837, A371920.

%K nonn,easy

%O 1,12

%A _Amiram Eldar_, Apr 12 2024