A359547 Numbers such that they are not divisible by p^p for any prime p, but for some k-th arithmetic derivative (k >= 1) of n such a factor exists.

15, 26, 35, 39, 45, 50, 51, 55, 63, 69, 74, 75, 86, 87, 90, 91, 95, 99, 102, 106, 110, 111, 115, 117, 119, 122, 123, 125, 133, 134, 141, 143, 146, 147, 153, 155, 158, 159, 169, 171, 175, 178, 183, 187, 190, 194, 195, 198, 203, 207, 210, 213, 215, 218, 219, 225, 226, 230, 234, 235, 245, 247, 249, 250

Offset: 1

Author

Antti Karttunen, Jan 05 2023

Keywords

• nonn

Examples

			15 = 3*5 is present, as although it itself is not in A100716, its arithmetic derivative 15' = 8 is there.
26 = 2*13 is present, as although neither 26 nor 26' = 15 are in A100716, its second derivative = 26'' = 15' = 8 is there.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, 4096 X 4096 pixel raster with origin (0, 0) in the upper left corner and black pixels at (x, y), indicate a number 4096*(y-1) + (x-1) in this sequence. Thus this image contains 7852685 terms of this sequence.

Crossrefs

Intersection of A048103 and A099309. Setwise difference A099309 \ A100716.

Cf. A003415, A327934 (subsequence), A359545, A359546 (characteristic function).

Programs

Mathematica

f[n_] := f[n] = Which[Abs@ n < 2, 0, PrimeQ[n], 1, True, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; g[n_] := And[n > 0, AnyTrue[FactorInteger[n], #2 >= #1 & @@ # &]]; w = {}; nn = 2^16; k = 1; While[Set[m, #^#] <= nn &[Prime[k]], AppendTo[w, m]; k++]; Reap[Do[If[! g[n], If[g@ NestWhile[f, n, And[! Divisible[#, 4], FreeQ[w, #]] &], Sow[n] ] ], {n, 2, nn}] ][[-1, -1]]
(* or, generate up to 7852685 terms of this sequence from the bitmap by setting y to a number not exceeding 4096: *)
With[{img = https://oeis.org/A359547/a359547.png, y = 2}, Map[4096 (#1 - 1) + #2 - 1 & @@ # &, Position[ImageData[img][[1 ;; y, All]], 0.]] ] (* Michael De Vlieger, Jan 23 2023 *)

PARI

isA359547(n) = A359546(n);