A378179 Squarefree numbers k whose neighbors are neither squarefree nor prime powers.

19, 51, 53, 55, 89, 91, 97, 149, 151, 161, 163, 197, 199, 233, 235, 241, 249, 251, 269, 271, 293, 295, 305, 307, 337, 339, 341, 349, 377, 379, 413, 415, 449, 451, 485, 487, 489, 491, 521, 523, 551, 557, 559, 577, 579, 593, 595, 629, 631, 638, 649, 651, 665, 667

Offset

1,1

Links

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

Example

Numbers k < 12 are not in the sequence since 12 is the smallest number in A126706.
13 and 15 are not in the sequence since 14 is squarefree.
17 is not in the sequence since 16 = 2^4.
a(1) = 19 since 19 is the smallest squarefree number such that 18 = 2*3^2 and 20 = 2*5^2 are not prime powers, but are divisible by a prime square.
a(2) = 51 since 50 = 2*5^2 and 52 = 2^2*13.
a(3) = 53 since 54 = 2*3^3.
a(4) = 55 since 56 = 7*2^3.
a(5) = 89 since 88 = 2^3*11 and 90 = 2*3^2*5, etc.

Mathematica

Reap[Do[If[
  And[SquareFreeQ[n],
    AllTrue[n + {-1, 1}, Nor[SquareFreeQ[#], PrimePowerQ[#]] &]],
  Sow[n]], {n, 1000}] ][[-1, 1]]

Prog

(PARI) list(lim)=my(v=List(), l1, l2); forfactored(k=18, lim\1+1, if(!issquarefree(k) && !issquarefree(l2) && issquarefree(l1) && #k[2][, 1]>1 && #l2[2][, 1]>1, listput(v, l1[1])); l2=l1; l1=k); Vec(v) \\ Charles R Greathouse IV, Nov 27 2024

Crossrefs

Subsequence of A073247.

Cf. A005117, A013929, A024619, A126706.

Sequence in context: A156376 A063306 A267572 * A349554 A190267 A373008

Adjacent sequences: A378176 A378177 A378178 * A378180 A378181 A378182

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 24 2024

Status

approved