A372399 Numbers k such that A124652(k) does not divide A372111(k-1).

2, 4, 6, 8, 9, 10, 14, 19, 21, 23, 25, 32, 34, 35, 36, 37, 38, 39, 45, 47, 48, 52, 54, 56, 57, 61, 65, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 94, 96, 97, 99, 100, 106, 108, 110, 113, 114, 122, 123, 130, 136, 142, 153, 157, 158, 159, 170, 171

Offset: 1

Author

Michael De Vlieger, May 05 2024

Keywords

• nonn

Comments

Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.

For n > 2, 1 < gcd(b(a(n)), s(a(n)-1)) < b(a(n)).

For n > 2, both b(a(n)) and s(a(n)-1) are necessarily composite, since prime p either divides or is coprime to n. Furthermore, both b(a(n)) and s(a(n)-1) have at least 2 distinct prime factors.

Indices of records in A124652 except {1, 2, 3, 5} are in this sequence.

Examples

			a(1) = 2 since b(2) = 2 does not divide s(1) = 1.
a(2) = 4 since b(4) = 4 does not divide s(3) = 6.
a(3) = 6 since b(6) = 9 does not divide s(5) = 15.
a(4) = 8 since b(8) = 8 does not divide s(7) = 30.
a(5) = 9 since b(9) = 16 does not divide s(8) = 38, etc.
Table of b(k) and s(k-1), where k = a(n), n = 2..12. Asterisked k denote terms such that rad(b(k)) | rad(s(k-1)); k = 73 and k = 4316 are the only other known indices where the terms have this quality.
     k      b(k)                        s(k-1)
    ----------------------------------------------------------
     4      4 =  2^2                    6 =  2 * 3
     6      9 =  3^2                   15 =  3 * 5
     8      8 =  2^3                   30 =  2 * 3 * 5
     9     16 =  2^4                   38 =  2 * 19
    10*    12 =  2^2 * 3               54 =  2 * 3^3
    14*    28 =  2^2 * 7               98 =  2 * 7^2
    19     32 =  2^5                  216 =  2^3 * 3^3
    21     81 =  3^4                  279 =  3^2 * 31
    23     20 =  2^2 * 5              370 =  2 * 5 * 37
    25    169 = 13^2                  403 = 13 * 31
    32     49 =  7^2                  728 =  2^3 * 7 * 13
    ...
    73*   100 =  2^2 * 5^2           4800 =  2^6 * 3 * 5^2
    ...
  4316*  4720 =  2^4 * 5 * 59    30806850 =  2 * 3 * 5^2 * 59^3
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of A124652(n), n = 1..10^5, showing A124652(a(n)) in red.

Crossrefs

Cf. A007947, A124652, A272618, A372111.

Programs

Mathematica

nn = 120; 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];
{2}~Join~Reap[Do[
  r = f[s]; k = SelectFirst[r, ! c[#] &];
  If[! Divisible[s, k], Sow[i]];
  c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

Formula

A124652(a(n)) is a number in row A372111(a(n)-1) of A272618.