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A381750 Nonprime-powers k such that, for any prime p dividing k and m = 1+floor(log k/log p), rad(p^m (mod k)) divides k, where rad = A007947.

%I A381750 #23 May 30 2025 23:15:45
%S A381750 6,12,14,24,39,56,62,112,120,155,254,992,1984,3279,5219,16256,16382,
%T A381750 19607,32512,70643,97655,208919,262142,363023,402233,712979,1040603,
%U A381750 1048574,1508597,2265383,2391483,4685519,5207819,6728903,21243689,25239899,56328959,61035155,67100672
%N A381750 Nonprime-powers k such that, for any prime p dividing k and m = 1+floor(log k/log p), rad(p^m (mod k)) divides k, where rad = A007947.
%C A381750 The number p^m is the smallest power of p dividing k that exceeds k, where m = 1+floor(log k/log p).
%C A381750 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 A381750 Prime powers k = p^m, m >= 0 have omega(k) = 1 and all r in S(n,p) are such that rad(r) | n.
%C A381750 Numbers k in this sequence have omega(k) > 1 and all r in S(n,p) are such that rad(r) | n.
%C A381750 A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).
%C A381750 Intersection of this sequence and A381525 is {6}.
%C A381750 Row a(n) of A381799 only contains powers of primes, i.e., row a(n) of A381799 is a proper subset of A000961.
%e A381750 Table of a(n) for n = 1..10, showing prime decomposition (facs(a(n))), and S(n,p_x), where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc.
%e A381750                          Numbers in row n of A381799:
%e A381750  n   a(n)  facs(a(n))    S(n,p_1)            S(n,p_2)        S(n,p_3)
%e A381750 ---------------------------------------------------------------------
%e A381750  1     6   2 * 3         {1,2,4},            {1,3}
%e A381750  2    12   2^2 * 3       {1,2,4,8},          {1,3,9}
%e A381750  3    14   2 * 7         {1,2,4,8},          {1,7}
%e A381750  4    24   2^3 * 3       {1,2,4,8,16},       {1,3,9}
%e A381750  5    39   3 * 13        {1,3,9,27},         {1,13}
%e A381750  6    56   2^3 * 7       {1,2,4,8,16,32},    {1,7,49}
%e A381750  7    62   2 * 31        {1,2,4,8,16,32},    {1,31}
%e A381750  8   112   2^4 * 7       {1,2,4,8,16,32,64}, {1,7,49}
%e A381750  9   120   2^3 * 3 * 5   {1,2,4,8,16,32,64}, {1,3,9,27,81}, {1,5,25}
%e A381750 10   155   5 * 31        {1,5,25,125},       {1,31}
%e A381750 .
%e A381750 a(1) = 6, the smallest number that is not a prime power, since 2^3 mod 6 = 2, and 3^2 mod 6 = 3, both divide 6.
%e A381750 10 is not in the sequence since 2^4 mod 10 = 6, rad(6) does not divide 10.
%e A381750 a(2) = 12 since 2^4 mod 12 = 4, rad(4) | 12, and 3^3 mod 12 = 3, rad(3) | 12.
%e A381750 a(3) = 14 since 2^4 mod 14 = 2 and 7^2 mod 14 = 7, both divide 14.
%e A381750 15 is not in the sequence since 3^3 mod 15 = 12, rad(12) does not divide 15, etc.
%t A381750 nn = 10^5;
%t A381750 Monitor[Reap[Do[
%t A381750   If[! PrimePowerQ[n],
%t A381750     If[AllTrue[
%t A381750       Map[PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ],
%t A381750         Divisible[n, rad[#]] &],
%t A381750       Sow[n] ] ], {n, 2, nn}] ][[-1, 1]], n]
%Y A381750 Cf. A000961, A007947, A024619, A139257, A381525, A381799.
%K A381750 nonn
%O A381750 1,1
%A A381750 _Michael De Vlieger_, Mar 27 2025