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.

6, 12, 14, 24, 39, 56, 62, 112, 120, 155, 254, 992, 1984, 3279, 5219, 16256, 16382, 19607, 32512, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67100672

Offset: 1

Author

Michael De Vlieger, Mar 27 2025

Keywords

• nonn

Comments

The number p^m is the smallest power of p dividing k that exceeds k, where m = 1+floor(log k/log p).

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}.

Prime powers k = p^m, m >= 0 have omega(k) = 1 and all r in S(n,p) are such that rad(r) | n.

Numbers k in this sequence have omega(k) > 1 and all r in S(n,p) are such that rad(r) | n.

A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).

Intersection of this sequence and A381525 is {6}.

Row a(n) of A381799 only contains powers of primes, i.e., row a(n) of A381799 is a proper subset of A000961.

Examples

			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.
                         Numbers in row n of A381799:
 n   a(n)  facs(a(n))    S(n,p_1)            S(n,p_2)        S(n,p_3)
---------------------------------------------------------------------
 1     6   2 * 3         {1,2,4},            {1,3}
 2    12   2^2 * 3       {1,2,4,8},          {1,3,9}
 3    14   2 * 7         {1,2,4,8},          {1,7}
 4    24   2^3 * 3       {1,2,4,8,16},       {1,3,9}
 5    39   3 * 13        {1,3,9,27},         {1,13}
 6    56   2^3 * 7       {1,2,4,8,16,32},    {1,7,49}
 7    62   2 * 31        {1,2,4,8,16,32},    {1,31}
 8   112   2^4 * 7       {1,2,4,8,16,32,64}, {1,7,49}
 9   120   2^3 * 3 * 5   {1,2,4,8,16,32,64}, {1,3,9,27,81}, {1,5,25}
10   155   5 * 31        {1,5,25,125},       {1,31}
.
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.
10 is not in the sequence since 2^4 mod 10 = 6, rad(6) does not divide 10.
a(2) = 12 since 2^4 mod 12 = 4, rad(4) | 12, and 3^3 mod 12 = 3, rad(3) | 12.
a(3) = 14 since 2^4 mod 14 = 2 and 7^2 mod 14 = 7, both divide 14.
15 is not in the sequence since 3^3 mod 15 = 12, rad(12) does not divide 15, etc.
		

Crossrefs

Cf. A000961, A007947, A024619, A139257, A381525, A381799.

Programs

Mathematica

nn = 10^5;
Monitor[Reap[Do[
  If[! PrimePowerQ[n],
    If[AllTrue[
      Map[PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ],
        Divisible[n, rad[#]] &],
      Sow[n] ] ], {n, 2, nn}] ][[-1, 1]], n]