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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.

%I A359547 #12 Jan 24 2023 02:57:06
%S A359547 15,26,35,39,45,50,51,55,63,69,74,75,86,87,90,91,95,99,102,106,110,
%T A359547 111,115,117,119,122,123,125,133,134,141,143,146,147,153,155,158,159,
%U A359547 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
%N 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.
%H A359547 Michael De Vlieger, <a href="/A359547/b359547.txt">Table of n, a(n) for n = 1..16384</a>
%H A359547 Michael De Vlieger, <a href="/A359547/a359547.png">4096 X 4096 pixel raster</a> 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.
%e A359547 15 = 3*5 is present, as although it itself is not in A100716, its arithmetic derivative 15' = 8 is there.
%e A359547 26 = 2*13 is present, as although neither 26 nor 26' = 15 are in A100716, its second derivative = 26'' = 15' = 8 is there.
%t A359547 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]]
%t A359547 (* or, generate up to 7852685 terms of this sequence from the bitmap by setting y to a number not exceeding 4096: *)
%t A359547 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 *)
%o A359547 (PARI) isA359547(n) = A359546(n);
%Y A359547 Intersection of A048103 and A099309. Setwise difference A099309 \ A100716.
%Y A359547 Cf. A003415, A327934 (subsequence), A359545, A359546 (characteristic function).
%K A359547 nonn
%O A359547 1,1
%A A359547 _Antti Karttunen_, Jan 05 2023