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

%I A372399 #11 Mar 29 2025 23:58:15
%S A372399 2,4,6,8,9,10,14,19,21,23,25,32,34,35,36,37,38,39,45,47,48,52,54,56,
%T A372399 57,61,65,74,75,76,77,81,82,83,84,85,86,87,88,89,90,91,94,96,97,99,
%U A372399 100,106,108,110,113,114,122,123,130,136,142,153,157,158,159,170,171
%N A372399 Numbers k such that A124652(k) does not divide A372111(k-1).
%C A372399 Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
%C A372399 For n > 2, 1 < gcd(b(a(n)), s(a(n)-1)) < b(a(n)).
%C A372399 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.
%C A372399 Indices of records in A124652 except {1, 2, 3, 5} are in this sequence.
%H A372399 Michael De Vlieger, <a href="/A372399/b372399.txt">Table of n, a(n) for n = 1..10000</a>
%H A372399 Michael De Vlieger, <a href="/A372399/a372399.png">Log log scatterplot of A124652(n)</a>, n = 1..10^5, showing A124652(a(n)) in red.
%F A372399 A124652(a(n)) is a number in row A372111(a(n)-1) of A272618.
%e A372399 a(1) = 2 since b(2) = 2 does not divide s(1) = 1.
%e A372399 a(2) = 4 since b(4) = 4 does not divide s(3) = 6.
%e A372399 a(3) = 6 since b(6) = 9 does not divide s(5) = 15.
%e A372399 a(4) = 8 since b(8) = 8 does not divide s(7) = 30.
%e A372399 a(5) = 9 since b(9) = 16 does not divide s(8) = 38, etc.
%e A372399 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.
%e A372399      k      b(k)                        s(k-1)
%e A372399     ----------------------------------------------------------
%e A372399      4      4 =  2^2                    6 =  2 * 3
%e A372399      6      9 =  3^2                   15 =  3 * 5
%e A372399      8      8 =  2^3                   30 =  2 * 3 * 5
%e A372399      9     16 =  2^4                   38 =  2 * 19
%e A372399     10*    12 =  2^2 * 3               54 =  2 * 3^3
%e A372399     14*    28 =  2^2 * 7               98 =  2 * 7^2
%e A372399     19     32 =  2^5                  216 =  2^3 * 3^3
%e A372399     21     81 =  3^4                  279 =  3^2 * 31
%e A372399     23     20 =  2^2 * 5              370 =  2 * 5 * 37
%e A372399     25    169 = 13^2                  403 = 13 * 31
%e A372399     32     49 =  7^2                  728 =  2^3 * 7 * 13
%e A372399     ...
%e A372399     73*   100 =  2^2 * 5^2           4800 =  2^6 * 3 * 5^2
%e A372399     ...
%e A372399   4316*  4720 =  2^4 * 5 * 59    30806850 =  2 * 3 * 5^2 * 59^3
%t A372399 nn = 120; c[_] := False;
%t A372399 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A372399 f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
%t A372399 Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
%t A372399 {2}~Join~Reap[Do[
%t A372399   r = f[s]; k = SelectFirst[r, ! c[#] &];
%t A372399   If[! Divisible[s, k], Sow[i]];
%t A372399   c[k] = True;
%t A372399   s += k, {i, 3, nn}] ][[-1, 1]]
%Y A372399 Cf. A007947, A124652, A272618, A372111.
%K A372399 nonn
%O A372399 1,1
%A A372399 _Michael De Vlieger_, May 05 2024