A051953 -id:A051953

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A053475

1 + the number of iterations of A051953 (Euler-cototient) function needed to reach 0, starting at n.

+20
11

2, 3, 3, 4, 3, 5, 3, 5, 4, 6, 3, 6, 3, 6, 4, 6, 3, 7, 3, 7, 5, 7, 3, 7, 4, 7, 5, 7, 3, 8, 3, 7, 4, 8, 4, 8, 3, 8, 5, 8, 3, 9, 3, 8, 6, 8, 3, 8, 4, 9, 4, 8, 3, 9, 5, 8, 6, 9, 3, 9, 3, 8, 6, 8, 4, 9, 3, 9, 5, 9, 3, 9, 3, 9, 5, 9, 4, 10, 3, 9, 6, 10, 3, 10, 6, 9, 4, 9, 3, 10, 4, 9, 5, 9, 4, 9, 3, 9, 6, 10, 3, 10

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

OFFSET

1,1

COMMENTS

Analogous sequences of iteration-lengths for A000005 or A000010 are A036459 and A049108 resp. The length values of 3 occur if the initial value is prime resulting in {p,1,0} iterations.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A076640(n) + 1. - Michael De Vlieger, Jul 04 2016

EXAMPLE

Starting with n=18, the iterations of A051953 are as follows: {18,12,8,4,2,1,0}. The length of this sequence is 7, so a(18) = 7. The function is applied a(n)-1 times.

MATHEMATICA

Table[Length@ NestWhileList[# - EulerPhi@ # &, n, # > 0 &], {n, 84}] (* Michael De Vlieger, Jul 04 2016 *)

CROSSREFS

Cf. A000005, A000010, A051953, A036459, A049108, A076640.

KEYWORD

nonn

AUTHOR

Labos Elemer, Jan 14 2000

STATUS

approved

A323410

Unitary analog of cototient function A051953: a(n) = n - A047994(n).

+20
11

0, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 6, 1, 8, 7, 1, 1, 10, 1, 8, 9, 12, 1, 10, 1, 14, 1, 10, 1, 22, 1, 1, 13, 18, 11, 12, 1, 20, 15, 12, 1, 30, 1, 14, 13, 24, 1, 18, 1, 26, 19, 16, 1, 28, 15, 14, 21, 30, 1, 36, 1, 32, 15, 1, 17, 46, 1, 20, 25, 46, 1, 16, 1, 38, 27, 22, 17, 54, 1, 20, 1, 42, 1, 48, 21, 44, 31, 18, 1, 58, 19, 26, 33, 48

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

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..21210

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

FORMULA

a(n) = n - A047994(n), where A047994 is unitary phi.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065463 = 0.2955577... . - Amiram Eldar, Dec 15 2023

MATHEMATICA

a[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 08 2023 *)

PROG

(PARI)

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };

A323410(n) = (n-A047994(n));

CROSSREFS

Cf. A047994, A065463.

Cf. also A034460, A051953, A323409, A323413.

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Jan 15 2019

STATUS

approved

A063985

Partial sums of cototient sequence A051953.

+20
10

0, 1, 2, 4, 5, 9, 10, 14, 17, 23, 24, 32, 33, 41, 48, 56, 57, 69, 70, 82, 91, 103, 104, 120, 125, 139, 148, 164, 165, 187, 188, 204, 217, 235, 246, 270, 271, 291, 306, 330, 331, 361, 362, 386, 407, 431, 432, 464, 471, 501, 520, 548, 549, 585, 600, 632, 653, 683

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

OFFSET

1,3

COMMENTS

Number of elements in the set {(x,y): 1 <= x <= y <= n, 1 = gcd(x,y)}; a(n) = A000217(n) - A002088(n) = A100613(n) - A185670(n). - Reinhard Zumkeller, Jan 21 2013

8*a(n) is the number of dots not in direct reach via a straight line from the center of a 2*n+1 X 2*n+1 array of dots. - Kiran Ananthpur Bacche, May 25 2022

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

FORMULA

a(n) = Sum_{x=1..n} (x - phi(x)) = Sum(x) - Sum(phi(x)) = A000217(n) - A002088(n), phi(n) = A000010(n), cototient(n) = A051953(n).

a(n) = n^2 - A091369(n). - Enrique Pérez Herrero, Feb 25 2012

G.f.: x/(1 - x)^3 - (1/(1 - x))*Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 18 2017

a(n) = (1/2 - 3/Pi^2)*n^2 + O(n*log(n)). - Amiram Eldar, Jul 26 2022

MATHEMATICA

f[n_] := n(n + 1)/2 - Sum[ EulerPhi@i, {i, n}]; Array[f, 58] (* Robert G. Wilson v *)

Accumulate[Table[n-EulerPhi[n], {n, 1, 60}]] (* Harvey P. Dale, Aug 19 2015 *)

PROG

(PARI) { a=0; for (n=1, 1000, write("b063985.txt", n, " ", a+=n - eulerphi(n)) ) } \\ Harry J. Smith, Sep 04 2009

(Haskell)

a063985 n = length [()| x <- [1..n], y <- [x..n], gcd x y > 1]

-- Reinhard Zumkeller, Jan 21 2013

(Python)

from sympy.ntheory import totient

def a(n): return sum(x - totient(x) for x in range(1, n + 1))

[a(n) for n in range(1, 51)] # Indranil Ghosh, Mar 18 2017

(Python)

from functools import lru_cache

@lru_cache(maxsize=None)

def A063985(n): # based on second formula in A018805

if n == 0:

return 0

c, j = 0, 2

k1 = n//j

while k1 > 1:

j2 = n//k1 + 1

c += (j2-j)*(k1*(k1+1)-2*A063985(k1)-1)

j, k1 = j2, n//j2

return (2*n+c-j)//2 # Chai Wah Wu, Mar 24 2021

(Java)

// Save the file as A063985.java to compile and run

import java.util.stream.IntStream;

import java.util.*;

public class A063985 {

public static int getInvisiblePoints(int n) {

Set<Float> slopes = new HashSet<Float>();

IntStream.rangeClosed(1, n).forEach(i ->

{IntStream.rangeClosed(1, n).forEach(j ->

slopes.add(Float.valueOf((float)i/(float)j))); });

return (n * n - slopes.size() + n - 1) / 2;

}

public static void main(String args[]) throws Exception {

IntStream.rangeClosed(1, 30).forEach(i ->

System.out.println(getInvisiblePoints(i)));

}

} // Kiran Ananthpur Bacche, May 25 2022

CROSSREFS

Cf. A000010, A000217, A002088, A048290, A051953.

KEYWORD

nonn

AUTHOR

Labos Elemer, Sep 06 2001

EXTENSIONS

Corrected by Robert G. Wilson v, Dec 13 2006

STATUS

approved

A065385

Numbers m at which value of cototient function (A051953) reaches a new record: cototient(m) > cototient(k) for all k < m.

+20
8

1, 2, 4, 6, 10, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 144, 150, 168, 180, 198, 204, 210, 240, 252, 264, 270, 294, 300, 330, 360, 378, 390, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600, 630, 660, 690, 714

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

OFFSET

1,2

COMMENTS

For totient values prime numbers give similar records.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

FORMULA

a=0; s=0; Do[s=n-EulerPhi[n]; If[s>a, a=s; Print[n]], {n, 1, 10000}]

EXAMPLE

a(8) = 30 because for m = 1...29 the cototient values are all smaller than cototient(30) = 22 = A065386(8) and this is the 8th number at which such a record is reached; analogous sequences are A002093, A002182, A015702 or A005250 for functions other than cototient.

MATHEMATICA

With[{s = Array[# - EulerPhi@ # &, 10^3]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, May 16 2018 *)

PROG

(PARI) r=-1; for(n=1, 1000, d=n-eulerphi(n); if(r<d, r=d; print1(n, ", ")))

(PARI) { n=0; x=-1; for (m=1, 10^9, c=m - eulerphi(m); if (c > x, x=c; write("b065385.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 17 2009

CROSSREFS

Cf. A051953, A065386.

KEYWORD

nonn

AUTHOR

Labos Elemer, Nov 05 2001

STATUS

approved

A319700

a(n) = A051953(A252463(n)).

+20
8

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 4, 1, 3, 1, 6, 6, 1, 1, 8, 3, 1, 4, 8, 1, 7, 1, 8, 8, 1, 7, 12, 1, 1, 12, 12, 1, 9, 1, 12, 8, 1, 1, 16, 5, 5, 14, 14, 1, 9, 9, 16, 18, 1, 1, 22, 1, 1, 12, 16, 13, 13, 1, 18, 20, 11, 1, 24, 1, 1, 12, 20, 11, 15, 1, 24, 8, 1, 1, 30, 15, 1, 24, 24, 1, 21, 15, 24, 30, 1, 19, 32, 1, 7, 16, 30, 1, 19, 1, 28, 22

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

OFFSET

1,8

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(n) = A051953(A252463(n)).

PROG

(PARI)

A051953(n) = (n - eulerphi(n));

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A252463(n) = if(!(n%2), n/2, A064989(n));

A319700(n) = A051953(A252463(n));

CROSSREFS

Cf. A051953, A252463.

Cf. also A319699, A319703, A319989, A320107, A320111.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 22 2018

STATUS

approved

A290087

a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

+20
7

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 5, 4, 5, 1, 5, 1, 5, 4, 5, 1, 8, 3, 4, 4, 8, 1, 6, 1, 8, 7, 4, 6, 13, 1, 8, 8, 13, 1, 8, 1, 13, 11, 13, 1, 17, 4, 8, 10, 11, 1, 11, 8, 17, 11, 8, 1, 18, 1, 17, 10, 17, 9, 12, 1, 11, 14, 12, 1, 21, 1, 10, 19, 21, 9, 10, 1, 21, 10, 11, 1, 21, 11, 18, 16, 21, 1, 18, 10, 21, 18, 21, 12, 25, 1, 28, 19, 21, 1, 19, 1, 28, 29

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

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(1) = 0; for n > 1, a(n) = A289626(A051953(n)).

CROSSREFS

Cf. A051953, A054571, A289626, A290086, A290085, A290086, A290088, A290089.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 07 2017

STATUS

approved

A295885

Filter combining sum of proper divisors (A001065) and cototient (A051953) of n.

+20
7

1, 1, 1, 2, 1, 3, 1, 4, 5, 6, 1, 7, 1, 8, 9, 10, 1, 11, 1, 12, 13, 14, 1, 15, 16, 17, 18, 19, 1, 20, 1, 21, 22, 23, 24, 25, 1, 26, 27, 28, 1, 29, 1, 30, 31, 32, 1, 33, 34, 35, 36, 37, 1, 38, 27, 39, 40, 41, 1, 42, 1, 43, 44, 45, 46, 47, 1, 48, 49, 50, 1, 51, 1, 52, 53, 54, 46, 55, 1, 56, 57, 58, 1, 59, 40, 60, 61, 62, 1, 63, 36

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

OFFSET

1,4

LINKS

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

FORMULA

Restricted growth sequence transform of a(n) = (1/2)*(2 + ((A051953(n) + A001065(n))^2) - A051953(n) - 3*A001065(n)).

PROG

(PARI)

allocatemem(2^30);

up_to = 65537;

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };

write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }

A051953(n) = (n-eulerphi(n));

A001065(n) = (sigma(n)-n);

Anotsubmitted5(n) = (1/2)*(2 + ((A051953(n)+A001065(n))^2) - A051953(n) - 3*A001065(n));

write_to_bfile(1, rgs_transform(vector(up_to, n, Anotsubmitted5(n))), "b295885.txt");

CROSSREFS

Cf. A001065, A051953, A291765.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 03 2017

STATUS

approved

A300232

Restricted growth sequence transform of A286152, filter combining A051953(n) and A046523(n), cototient and the prime signature of n.

+20
7

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 27, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 47, 2, 50, 2, 51, 52, 53, 46, 54, 2, 55, 56, 57, 2, 58, 40, 59, 60, 61, 2, 62, 36, 63, 64, 65, 66, 67, 2, 68, 69

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

OFFSET

1,2

LINKS

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

EXAMPLE

a(39) = a(55) (= 27) because both are nonsquare semiprimes (3*13 and 5*11), and both have cototient value 15 = 39 - phi(39) = 55 - phi(55).

PROG

(PARI)

up_to = 65537;

write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }

A051953(n) = (n - eulerphi(n));

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011

A286152(n) = (2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n))/2;

write_to_bfile(1, rgs_transform(vector(up_to, n, A286152(n))), "b300232.txt");

CROSSREFS

Cf. A046523, A051953, A286152.

Cf. also A295885, A300223, A300224, A300226, A300229, A300230, A300231, A300233, A300235.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 01 2018

STATUS

approved

A300233

Filter sequence combining A051953(n) and A009194(n), cototient of n and gcd(n,sigma(n)).

+20
7

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 13, 2, 15, 16, 17, 14, 18, 2, 19, 2, 20, 21, 22, 23, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 26, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 45, 54, 2, 55, 43, 56, 2, 57, 39, 58, 59, 60, 2, 61, 62, 60, 63, 55, 64, 65, 2, 66

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

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of P(A051953(n), A009194(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

LINKS

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

EXAMPLE

a(20) = a(22) (= 13) because A051953(20) = A051953(22) = 12 and A009194(20) = A009194(22) = 2.

PROG

(PARI)

write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }

A009194(n) = gcd(n, sigma(n));

A051953(n) = (n - eulerphi(n));

Aux300233(n) = (1/2)*(2 + ((A051953(n)+A009194(n))^2) - A051953(n) - 3*A009194(n));

write_to_bfile(1, rgs_transform(vector(65537, n, Aux300233(n))), "b300233.txt");

CROSSREFS

Cf. A009194, A051953.

Cf. also A295885, A300223, A300226, A300229, A300230, A300231, A300232, A300235.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 01 2018

STATUS

approved

A049586

a(n) is the GCD of the cototients (A051953) of n and n+1.

+20
6

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3

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

OFFSET

0,9

COMMENTS

Most of the terms are 1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

FORMULA

a(n) = gcd(n + 1 - phi(n+1), n - phi(n)).

a(n) = gcd(A051953(n+1), A051953(n)).

MATHEMATICA

Map[GCD @@ Map[# - EulerPhi@ # &, #] &, Partition[Range@ 106, 2, 1]] (* Michael De Vlieger, Aug 09 2017 *)

PROG

(Magma) [GCD((n+1-EulerPhi(n+1)), n-EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Aug 10 2017

CROSSREFS

Cf. A000010, A051953, A058515.

KEYWORD

nonn

AUTHOR

Labos Elemer, Dec 28 2000

STATUS

approved

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