A371910 Position of A109890(n) among the sorted set of divisors of A109735(n-1).

2, 4, 4, 4, 7, 6, 3, 4, 9, 5, 6, 12, 9, 9, 11, 14, 9, 13, 9, 4, 4, 3, 6, 7, 6, 10, 12, 5, 5, 6, 8, 9, 13, 12, 4, 15, 5, 3, 4, 6, 8, 4, 9, 17, 7, 2, 5, 3, 8, 7, 6, 13, 8, 17, 6, 7, 4, 9, 10, 8, 13, 17, 15, 7, 3, 7, 13, 5, 6, 16, 8, 11, 8, 5, 4, 13, 12, 17, 5, 6

Offset: 3

Author

Michael De Vlieger, Apr 26 2024

Keywords

• nonn

Comments

A109890(n) is the a(n)-th smallest divisor of A109735(n-1).

Examples

			Table relating sequences b = A109890, s = A109735, c = A371909. a(n) = c(n) implies both A111315(i) = n and A111316(i) = b(n) = s(n-1).
    n     b(n)   s(n-1)  a(n)  c(n)    i
   --------------------------------------
    3       3   =    3     2     2     1
    4       6   =    6     4     4     2
    5       4       12     4     6
    6       8       16     4     5
    7      12       24     7     8
    8       9       36     6     9
    9       5       45     3     6
   10      10       50     4     6
   11      15       60     9    12
   12      25       75     5     6
  ...
  222  113573 = 113573     4     4     3
  ...
  232  230801 = 230801     4     4     4
  ...
  279  941071 = 941071     4     4     5
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A000005, A027750, A109735, A109890, A111315, A111316, A371909.

Programs

Mathematica

nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[d = Divisors[s]; k = SelectFirst[d, ! c[#] &];
    c[k] = True; Sow[FirstPosition[d, k][[1]]];
    s += k, {n, 3, nn}] ][[-1, 1]]

Formula

1 < a(n) <= A371909(n), where A371909(n) = A000005(A109735(n-1)), corollary of Sloane's theorem in the comments in A109890.

A109890(n) = T(j, k), where T = A027750, j = A109735(n-1), and k = a(n).

A371909(n) = A371910(n) if and only if A109890(n) = A109735(n-1).