A359546 -id:A359546

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A359550

Multiplicative with a(p^e) = 1 if p > e, otherwise 0.

+10
28

1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions.

FORMULA

Multiplicative with a(p^e) = [e < p], where [ ] is the Iverson bracket.

a(n) = [A129251(n) == 0] = [A327936(n) == 1] = [A342007(n) == 1].

a(n) = 1 - A342023(n).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 - 1/p^p) = 0.7219902344... . - Amiram Eldar, Jan 07 2023

MATHEMATICA

f[p_, e_] := If[e < p, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 06 2023 *)

PROG

(PARI) A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

(Python)

from sympy import factorint

def A359550(n): return int(all(map(lambda d:d[0]>d[1], factorint(n).items()))) # Chai Wah Wu, Jan 06 2023

CROSSREFS

Characteristic function of A048103.

Cf. A129251, A327936, A328308, A342007, A342023 (one's complement), A359546, A359551 (Dirichlet inverse).

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Jan 06 2023

STATUS

approved

A328308

a(n) = 1 if k-th arithmetic derivative of n is zero for some k, otherwise 0.

+10
16

1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Question: What is the asymptotic mean of this and related sequences like A368915? Compare also to A341996, A359543 and A359546.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Michael De Vlieger, Bitmap of a(n), n = 0..2^24, 2048 X 2048 pixels, with 0 in white and 1 in black. Furnishes 4260302 terms of A099308.

Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seq., Vol. 6 (2003), Article 03.3.4.

Index entries for characteristic functions

FORMULA

For prime p, a(p) = 1, a(p^p * m) = 0, for all m >= 1. a(4m) = 0 for m > 0. - Michael De Vlieger, Jan 04 2023

From Antti Karttunen, Jan 06 2023: (Start)

a(0) = 1; and for n > 0, a(n) = A359550(n) * a(A003415(n)). [Provided that Conjecture 3 given on page 5 of Ufnarovski and Åhlander 2003 paper holds, i.e., that map x -> x' never forms nontrivial loops]

a(n) = 1 - A341999(n).

a(n) >= A359543(n).

(End)

For all n > 1, a(n) <= A368915(n) <= A359550(n). - Antti Karttunen, Jan 10 2024

MATHEMATICA

w = {}; nn = 2^10; k = 1; While[Set[m, #^#] <= nn &[Prime[k]], AppendTo[w, m]; k++]; a3415[n_] := a3415[n] = Which[Abs@ n < 2, 0, PrimeQ[n], 1, True, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]{1, 1}~Join~Reap[Do[Which[PrimeQ[n], Sow[1], MemberQ[w, n], Sow[0], True, If[NestWhileList[a3415, n, And[! Divisible[#, 4], FreeQ[w, #]] &, 1][[-1]] == 0, Sow[1], Sow[0]]], {n, 2, nn}]][[-1, -1]] (* Michael De Vlieger, Jan 04 2023 *)

(* 2nd program: generate m <= 2^24 terms of the sequence from the bitmap above: *)

m = 10^3; Flatten[ImageData[Import["https://oeis.org/A328308/a328308.png"], "Bit"]][[1 ;; m]] /. {0 -> 1, 1 -> 0} (* Michael De Vlieger, Jan 04 2023 *)

PROG

(PARI)

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));

A328308(n) = if(!n, 1, while(n>1, n = A003415checked(n)); (n));

CROSSREFS

Characteristic function of A099308.

Cf. A003415, A099309 (positions of zeros), A256750, A328306 [= a(A276086(n))], A328309 (partial sums), A341996, A341999 (one's complement), A342023, A351071, A359541 (inverse Möbius transform), A359543, A359546, A359550, A368915.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 12 2019

STATUS

approved

A341999

a(n) = 1 if the k-th arithmetic derivative is nonzero for all k >= 0, otherwise 0.

+10
13

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Characteristic function of A099309.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for characteristic functions

FORMULA

a(n) = 1 if n is in A100716 or ends there by repeated applications of A003415, otherwise a(n) = 0 (when n instead reaches 0 by such iteration).

For all n, a(n) >= A341996(n).

For all n > 0, a(A099309(n)) = a(A100716(n)) = 1.

For all n > 0, a(n) = [A256750(n) < 1].

For all n > 0, a(n) >= [A129251(n)>0], i.e., if A129251(n) is nonzero, then certainly a(n) = 1.

For all n > 1, a(n) >= [A341997(n) > 1].

a(n) = 1 - A328308(n), and for n >= 1, a(n) = A342023(n) + A359546(n). - Antti Karttunen, Jan 05 2023

PROG

(PARI)

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));

A341999(n) = if(!n, n, while(n>1, n = A003415checked(n)); (!n));

CROSSREFS

Cf. A003415, A100716, A129251, A256750, A341996, A341997.

Cf. A099308 (positions of zeros), A099309 (of ones), A328308 (one's complement), A342023, A359542 (inverse Möbius transform), A359546.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 28 2021

STATUS

approved

A342023

a(n) = 1 if there is a prime p such that p^p divides n, otherwise 0.

+10
8

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions.

FORMULA

a(n) = min(1, A129251(n)) = [A129251(n) > 0], where [ ] is the Iverson bracket.

a(n) = [A342007(n) > 1] = [A327936(n) > 1].

For all n >= 1,

a(n) <= A107078(n), i.e., a(n) = 1 => A107078(n) = 1.

a(n) <= A342024(n), i.e., a(n) = 1 => A342024(n) = 1.

a(n) <= A341999(n), i.e., a(n) = 1 => A341999(n) = 1.

A342004

For all n >= 2, a(A003415(n)) = A341996(n).

For all n >= 0, a(A276086(n)) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Product_{p prime} (1 - 1/p^p) = 0.2780097655... . - Amiram Eldar, Jul 24 2022

a(n) = 1 - A359550(n) = A341999(n) - A359546(n). - Antti Karttunen, Jan 06 2023

MATHEMATICA

Array[Function[{D, q}, Boole[Total@ Table[Count[D, _?(IntegerExponent[#, p] == p &)], {p, Prime@ Range@ q}] > 0]] @@ {Divisors[#], PrimePi@ Floor[Sqrt[#]]} &, 120] (* Michael De Vlieger, Mar 11 2021 *)

PROG

(PARI) A342023(n) = if(1==n, 0, my(f = factor(n)); for(k=1, #f~, if(f[k, 2]>=f[k, 1], return(1))); (0));

(Python)

from sympy import factorint

def A342023(n):

f = factorint(n)

for p in f:

if p <= f[p]:

return 1

return 0 # Chai Wah Wu, Mar 09 2021

CROSSREFS

Characteristic function of A100716.

Cf. A003415, A008966, A048103 (positions of zeros), A107078, A276086, A341996, A341999, A327936, A341999, A342004, A342007, A359546, A359550 (one's complement).

Differs from A129251 and A276077 for the first time at n=108, as here a(108) = 1.

Differs from A342024 for the first time at n=625, where a(625)=0, while A342024(625)=1.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 09 2021

STATUS

approved

A359543

a(n) = 1 if A359542(n) = 0, otherwise a(n) = 0.

+10
5

1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Question: What can be said about the distribution of 0's and 1's in this sequence? Compare also to A328308 and A359546.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(n) = [A359542(n) == 0], where [ ] is the Iverson bracket.

a(n) = [A359541(n) == A000005(n)].

a(n) <= A328308(n).

PROG

(PARI)

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));

A341999(n) = if(!n, n, while(n>1, n = A003415checked(n)); (!n));

A359542(n) = sumdiv(n, d, A341999(d));

A359543(n) = (0==A359542(n));

CROSSREFS

Characteristic function of A359544.

Cf. A000005, A003415, A328308, A341999, A359541, A359542, A359546.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 05 2023

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

EXAMPLE

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.

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 *)

PROG

(PARI) isA359547(n) = A359546(n);

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 05 2023

STATUS

approved

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