A359545 -id:A359545 - cp's OEIS frontend

A099308 Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 30, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 105, 107, 109, 113, 114, 118, 121, 126, 127, 129, 130

Offset: 1

T. D. Noe, Oct 12 2004

nonn

The first derivative of 0 and 1 is 0. The second derivative of a prime number is 0.

For all n, A003415(a(n)) is also a term of the sequence. A351255 gives the nonzero terms as ordered by their position in A276086. - Antti Karttunen, Feb 14 2022

			18 is on this list because the first through fifth derivatives are 21, 10, 7, 1, 0.

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099309 (complement, numbers whose k-th arithmetic derivative is nonzero for all k), A351078 (first noncomposite reached when iterating the derivative from these numbers), A351079 (the largest term on such paths).
Cf. A328308, A328309 (characteristic function and their partial sums), A341999 (1 - charfun).
Cf. A276086, A328116, A351255 (permutation of nonzero terms), A351257, A351259, A351261, A351072 (number of prime(k)-smooth terms > 1).
Cf. also A256750 (number of iterations needed to reach either 0 or a number with a factor of the form p^p), A327969, A351088.
Union of A359544 and A359545.

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; d1=Table[dn[n], {n, 40000}]; nLim=200; lst={1}; i=1; While[i<=Length[lst], currN=lst[[i]]; pre=Intersection[Flatten[Position[d1, currN]], Range[nLim]]; pre=Complement[pre, lst]; lst=Join[lst, pre]; i++ ]; Union[lst]

\\ The following program would get stuck in nontrivial loops. However, we assume that the conjecture 3 in Ufnarovski & Åhlander paper holds ("The differential equation n^(k) = n has only trivial solutions p^p for primes p").
A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
isA099308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n)); \\ Antti Karttunen, Feb 14 2022

For all n >= 0, A328309(a(n)) = n. - Antti Karttunen, Feb 14 2022

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.

Original entry on oeis.org

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

Antti Karttunen, Jan 05 2023

nonn

			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.

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.

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

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

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);
```

A359544 Numbers k such that all their divisors (including k itself) are in A099308, i.e., reach eventually zero when iterated with the arithmetic derivative.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 29, 31, 33, 34, 37, 38, 41, 42, 43, 46, 47, 49, 53, 57, 58, 59, 61, 62, 65, 66, 67, 71, 73, 77, 79, 82, 83, 85, 89, 93, 94, 97, 98, 101, 103, 107, 109, 113, 114, 118, 121, 127, 129, 131, 137, 139, 142, 145, 149, 151, 154, 157, 161, 163, 166

Offset: 1

Antti Karttunen, Jan 05 2023

nonn

			30 = 2*3*5, although it is in A099308, is not included here because its divisor 15 is not in A099308.

Positions of 0's in A359542.
Setwise difference A099308 \ A359545.
Cf. A359543 (characteristic function).

PARI
```
isA359544(n) = A359543(n);
```

cp's OEIS Frontend

A099308 Numbers m whose k-th arithmetic derivative is zero for some k. Complement of A099309.

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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.

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A359544 Numbers k such that all their divisors (including k itself) are in A099308, i.e., reach eventually zero when iterated with the arithmetic derivative.

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Author

Keywords

Examples

Crossrefs

Programs

PARI