A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84

Offset

1,2

Comments

Numbers k whose reduced residue system does not intersect A286708 (i.e., powerful numbers that are not prime powers).

Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q^2 = A382767(k). Then this sequence is that of k such that k < m.

This sequence is finite following arguments akin to those in A051250 and A382659, with 626 terms.

Sequences A048597, A051250, and A382659 are proper subsets of this sequence.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..626

Example

Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = (2*3)^2 = 36.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 64, 81, 128}.
  31 is in the sequence since 31 < m, m = (2*3)^2 = 36, but 37 is not a term since 37 > 36.
  25 is in the sequence since 25 < m, m = 36.
  49 is not a term since 49 > 36, and 243 is not a term since 243 > 100, 100 = (2*5)^2, etc.
For omega = 2, we have the squarefree numbers {6, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218}.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, .., 1152}, since m = (5*7)^2 = 1225.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, ..., 400}, since m = (3*7)^2 = 441.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28, 56, ..., 224}, since m = (3*5)^2 = 225.
  Intersection with A033848 = {k : rad(k) = 15} is {15, 45, 75, 135}, since m = (2*7)^2 = 196, etc.

Mathematica

Select[Range[510510], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

Crossrefs

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A286708, A380539, A382659 (k such that k < p^2*q), A382767.

Sequence in context: A212554 A178772 A367141 * A296870 A085735 A269393

Adjacent sequences: A382957 A382958 A382959 * A382961 A382962 A382963

Keyword

nonn,easy,fini,full

Author

Michael De Vlieger, Apr 14 2025

Status

approved