A358089 First differences of A126706.

6, 2, 4, 4, 8, 4, 4, 1, 3, 2, 2, 2, 2, 4, 3, 5, 4, 3, 1, 4, 4, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 4, 2, 6, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 3, 1, 3, 1, 4, 4, 4, 1, 3, 4, 2, 2, 4, 3, 1, 4, 4, 4, 4, 1, 3, 4, 2, 2, 4, 2, 2, 1, 3, 2, 2, 8, 1, 3, 4, 2

Offset

1,1

Comments

A356322 relates to the first instances of exactly k consecutive 1's in this sequence.

a(n) - 1 = number of 0's between 1's in A355447.

For prime p, m such that m mod p^2, unless m = p^e, e > 1, is in A126706, as a consequence of definition of A126706. Therefore m <= 4 is common, m <= 9 much less so. Consequently, the arrangement of A126706 mod M for M in A061742 presents a quasi-modular pattern as seen in the example and raster link at A355447.

a(51265) = 7; m = 9 is not observed in the first 6577230 terms of the sequence, a dataset corresponding to terms k <= 2^24 in A126706.

Links

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

Example

The smallest numbers that are neither squarefree nor a prime power are {12, 18, 20, 24, 28 ...}, therefore the first terms of this sequence are {6, 2, 4, 4, ...}.

Mathematica

k = 0; Rest@ Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, Sow[n - k]; Set[k, n] ], {n, 270}] ][[-1, -1]]
(* Generate 317359 terms of this sequence from the image at A355447: *)
Differences@ Position[Flatten@ ImageData[Import["https://oeis.org/A355447/a355447_1.png", "PNG"]], 0.][[All, -1]]

Crossrefs

Cf. A126706, A355447, A356322.

Sequence in context: A173273 A084945 A370465 * A010493 A175286 A061496

Adjacent sequences: A358086 A358087 A358088 * A358090 A358091 A358092

Keyword

nonn,easy

Author

Michael De Vlieger, Oct 31 2022

Status

approved