A389374 Powers k^m, m > 1, where k is an Achilles number that has a primorial kernel but is not a product of primorials.

11664, 419904, 944784, 1259712, 3779136, 7290000, 15116544, 20250000, 76527504, 81000000, 136048896, 182250000, 262440000, 272097792, 306110016, 324000000, 544195584, 590490000, 918330048, 1224440064, 1296000000, 1640250000, 2025000000, 2361960000, 2754990144

Offset

1,1

Comments

Powers k^m, m > 1, where k is in A386433.

This sequence is A389226 \ A389260.

Proper subset of A369420 which in turn is a proper subset of A056808.

Powers k^m, m > 1, where k is an Achilles number such that A006530(k) < A053669(k), but is not a product of primorials.

Numbers N in A369420 whose exponents m of prime power factors prime(i)^m | N are such that m is not nondecreasing as prime index i increases.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Example

 n         a(n)
------------------------------------------
 1       11664 =   108^2   2^4 * 3^6
 2      419904 =   648^2   2^6 * 3^8
 3      944784 =   972^2   2^4 * 3^10
 4     1259712 =   108^3   2^6 * 3^9
 5     3779136 =  1944^2   2^6 * 3^10
 6     7290000 =  2700^2   2^4 * 3^6 * 5^4
 7    15116544 =  3888^2   2^8 * 3^10
 8    20250000 =  4500^2   2^4 * 3^4 * 5^6
 9    76527504 =  8748^2   2^4 * 3^14
10    81000000 =  9000^2   2^6 * 3^4 * 5^6
11   136048896 =   108^4   2^8 * 3^12
12   182250000 = 13500^2   2^4 * 3^6 * 5^6

Mathematica

nn = 2^32; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[#[[1, 1]] == 2, Length[#] > 1, Union@ Differences@ Map[PrimePi, #[[;; , 1]]] == {1}, Apply[GCD, #[[;; , -1]]] == 1, ReverseSort@ #[[;; , -1]] != #[[;; , -1]] ] &[FactorInteger[x]]; MapIndexed[Set[S[First[#2]], #1] &, Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[mm, 3]}, {a, Sqrt[mm/b^3]}], fQ]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]

Crossrefs

Cf. A001597, A001694, A002110, A006530, A007947, A025487, A052486, A053669, A056808, A055932, A126706, A286708, A383394, A386433, A389226, A389260.

Sequence in context: A214804 A352330 A389814 * A203821 A223598 A223439

Adjacent sequences: A389371 A389372 A389373 * A389375 A389376 A389377

Keyword

nonn

Author

Michael De Vlieger, Oct 10 2025

Status

approved