A394049 Smallest number k that is neither squarefree nor squareful such that rad(k) | s(n), where s = A120944 is the sequence of squarefree composite numbers and rad = A007947.

12, 20, 28, 45, 63, 44, 52, 12, 99, 68, 175, 76, 117, 12, 92, 153, 275, 171, 116, 124, 325, 12, 207, 20, 148, 539, 12, 164, 425, 172, 261, 637, 279, 188, 475, 12, 45, 212, 20, 333, 12, 575, 236, 833, 244, 369, 387, 20, 931, 268, 12, 423, 284, 1573, 725, 292, 28

Offset

1,1

Comments

If a(n) < s(n), then a(n) is in row s(n) of A162306.

If omega(a(n)) = 2, a(n) = A366807(n), where omega = A001221.

Terms have the ordered prime signature (2,1), that is, the product of the square of the smallest prime factor of A120944(n) and the second smallest prime factor of A120944(n). Thus, terms are in A096156.

Links

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

Formula

a(n) = A020639(A120944(n))^2 * A119288(A120944(n)).

Example

Table of n, s(n), a(n) for select n:
     n    s(n)                                 a(n)
  -------------------------------------------------
     1      6 = 2 * 3                2^2 *  3 = 12
     2     10 = 2 * 5                2^2 *  5 = 20
     3     14 = 2 * 7                2^2 *  7 = 28
     4     15 = 3 * 5                3^2 *  5 = 45
     5     21 = 3 * 7                3^2 *  7 = 63
     6     22 = 2 * 11               2^2 * 11 = 44
     7     26 = 2 * 13               2^2 * 13 = 52
     8     30 = 2 * 3 * 5            2^2 *  3 = 12
    82    210 = 2 * 3 * 5 * 7        2^2 *  3 = 12
  1061   2310 = 2 * 3 * 5 * 7 * 11   2^2 *  3 = 12

Mathematica

a120944Q[x_] := And[SquareFreeQ[x], CompositeQ[x]]; f[x_] := Apply[Times, FactorInteger[x][[;; 2, 1]]^{2, 1}]; Table[If[a120944Q[n], f[n], Nothing], {n, 160}]

Prog

(PARI) isfc(n) = issquarefree(n)&&!isprime(n)&&n>1; \\ A120944
f(n) = my(f=factor(n)); f[1, 1]^2*f[2, 1];
lista(nn) = apply(x->f(x), select(isfc, [1..nn])); \\ Michel Marcus, Apr 11 2026

Crossrefs

Cf. A001694, A005117, A013929, A020639, A052485, A096156, A119288, A120944, A126706, A162306, A332785, A388607.

Sequence in context: A110187 A096156 A364999 * A366807 A210968 A107277

Adjacent sequences: A394046 A394047 A394048 * A394050 A394051 A394052

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 08 2026

Status

approved