A371909 -id:A371909 - cp's OEIS frontend

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

Michael De Vlieger, Apr 26 2024

nonn

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

			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
  ...

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

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

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]]

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).

A372322 a(n) = A010846(A372111(n)).

Original entry on oeis.org

1, 2, 5, 6, 5, 11, 18, 8, 16, 22, 5, 28, 13, 33, 23, 38, 11, 26, 12, 9, 58, 28, 80, 5, 30, 55, 19, 27, 19, 56, 37, 21, 27, 87, 44, 44, 48, 38, 18, 58, 42, 5, 110, 26, 112, 140, 38, 45, 32, 144, 102, 59, 5, 139, 225, 39, 44, 22, 180, 86, 114, 34, 23, 133, 41, 115

Offset: 1

Michael De Vlieger, May 05 2024

nonn

Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
a(n) is the length of row A372111(n) of A162306.
Analogous to A371909, which instead regards A109890 and A109735.

			Let s(x) = A372111(x) and let r(x) = A010846(x).
a(1) = 1 since r(s(1)) = r(1) = 1.
a(2) = 2 since r(s(2)) = r(3) = 2. For prime p, r(p) = card({1, p}) = 2.
a(3) = 5 since r(s(3)) = r(6) = 5. r(6) = card({1, 2, 3, 4, 6}) = 5.
a(4) = 6 since r(s(4)) = r(10) = 6. r(10) = card({1, 2, 4, 5, 8, 10}) = 6.
a(5) = 5 since r(s(5)) = r(15) = 5. r(15) = card({1, 3, 5, 9, 15}) = 5.
a(6) = 11 since r(s(6)) = r(24) = 11. r(24) = card({1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24}) = 11, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A010846, A124652, A162306, A371909, A372111, A372323.

nn = 68; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{1}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  Sow[Length[r]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

cp's OEIS Frontend

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

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