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A382438 Numbers k in A024619 such that all residues r (mod k) in row k of A381801 are such that rad(r) divides k, where rad = A007947.

%I A382438 #36 May 31 2025 05:52:30
%S A382438 6,12,14,24,39,62,155,254,3279,5219,16382,19607,70643,97655,208919,
%T A382438 262142,363023,402233,712979,1040603,1048574,1508597,2265383,2391483,
%U A382438 4685519,5207819,6728903,21243689,25239899,56328959,61035155,67977559,150508643
%N A382438 Numbers k in A024619 such that all residues r (mod k) in row k of A381801 are such that rad(r) divides k, where rad = A007947.
%C A382438 Numbers k in A024619 such that A381804(k) = 0.
%C A382438 Let S(n,p) be the set of distinct power residues r (mod n) beginning with empty product and recursively multiplying by prime p | n. For example, S(10,2) = {1,2,4,8,6}.
%C A382438 This sequence builds on A381750, taking the tensor product T(k) (mod k) of S(k,p), p | k. If all products r (mod k) are such that rad(r) | k, then k is in this sequence. Distinct residues r (mod k) in T(k) are listed in row k of A381801.
%C A382438 Proper subset of A381750.
%C A382438 A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).
%C A382438 Conjecture: 12 and 24 are the only nonsquarefree numbers in this sequence, i.e., in A126706.
%H A382438 Michael De Vlieger, <a href="/A382438/a382438.txt">Efficient Mathematica code for this sequence</a> (2025).
%e A382438 Table of a(n) for n = 1..10, showing prime decomposition (facs(a(n))), row a(n) of A381801:
%e A382438                         Row a(n) of A381801
%e A382438  n    a(n)  facs(a(n))  k (mod a(n)) such that rad(k) | a(n).
%e A382438 -------------------------------------------------------------
%e A382438  1      6   2 * 3       {0, 1, 2, 3, 4}
%e A382438  2     12   2^2 * 3     {0, 1, 2, 3, 4, 6, 8, 9}
%e A382438  3     14   2 * 7       {0, 1, 2, 4, 7, 8}
%e A382438  4     24   2^3 * 3     {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
%e A382438  5     39   3 * 13      {0, 1, 3, 9, 13, 27}
%e A382438  6     62   2 * 31      {0, 1, 2, 4, 8, 16, 31, 32}
%e A382438  7    155   5 * 31      {0, 1, 5, 25, 31, 125}
%e A382438  8    254   2 * 127     {0, 1, 2, 4, 8, 16, 32, 64, 127, 128}
%e A382438  9   3279   3 * 1093    {0, 1, 3, 9, 27, 81, 243, 729, 1093, 2187}
%e A382438 10   5219   17 * 307    {0, 1, 17, 289, 307, 4913}
%e A382438 Let b = A381750.
%e A382438 a(1) = 6 since T(6) (mod 6) = {1,2,4} X {1,3} = {{1,2,4},{3,0,0}}; all residues r (mod 6) in T(6) (i.e., in row 6 of A381801) are such that rad(r) | 6.
%e A382438 a(2) = 12 since T(12) (mod 12) = {1,2,4,8} X {1,3,9} = {{1,2,4,8},{3,6,0,0},{9,6,0,0}}; all residues r (mod 12) in T(12) are such that rad(r) | 12.
%e A382438 a(3) = 14 since T(14) (mod 14) = {1,2,4,8} X {1,7} = {{1,2,4,8},{7,0,0,0}}; all residues r (mod 14) in T(14) are such that rad(r) | 14.
%e A382438 a(4) = 24 since T(24) (mod 24) = {1,2,4,8,16} X {1,3,9} = {{1,2,4,8,16},{3,6,12,0,0},{9,18,0,0,0}}; all residues r (mod 24) in T(24) are such that rad(r) | 24.
%e A382438 b(6) = 56 is not in the sequence since 49*2 = 98 = 42 (mod 56), rad(42) does not divide 56.
%e A382438 b(8) = 112 is not in the sequence since 49*4 = 196 = 84 (mod 112), rad(84) does not divide 112, etc.
%Y A382438 Cf. A007947, A024619, A381750, A381801.
%K A382438 nonn
%O A382438 1,1
%A A382438 _Michael De Vlieger_, Mar 27 2025