A081373 -id:A081373 - cp's OEIS frontend

A322873 Ordinal transform of A300721, which is Möbius transform of A060681.

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 1, 4, 3, 4, 1, 4, 1, 5, 2, 5, 1, 6, 2, 6, 2, 3, 1, 7, 1, 1, 2, 7, 2, 4, 1, 8, 3, 2, 1, 5, 1, 3, 3, 9, 1, 3, 2, 8, 4, 4, 1, 6, 2, 5, 3, 10, 1, 4, 1, 11, 1, 5, 3, 4, 1, 6, 2, 7, 1, 6, 1, 12, 2, 4, 1, 7, 1, 7, 4, 13, 1, 8, 1, 14, 2, 1, 1, 5, 2, 3, 3, 15, 1, 8, 1, 8, 2, 2, 1, 9, 1, 3, 4

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A060681, A300721, A081373, A303756, A322871, A322874.

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
A300721[n_] := Sum[MoebiusMu[n/d] A060681[d], {d, Divisors[n]}];
b[_] = 1;
a[n_] := a[n] = With[{t = A300721[n]}, b[t]++];
a /@ Range[1, 105] (* Jean-François Alcover, Dec 19 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A060681(n) = if(1==n,0,(n-(n/vecmin(factor(n)[, 1]))));
A300721(n) = sumdiv(n, d, moebius(n/d)*A060681(d));
v322873 = ordinal_transform(vector(up_to,n,A300721(n)));
A322873(n) = v322873[n];

A330747 Number of values of k, 1 <= k <= n, with A049559(k) = A049559(n), where A049559(n) = gcd(n - 1, phi(n)).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 7, 1, 8, 3, 9, 1, 10, 1, 11, 2, 12, 1, 13, 3, 14, 4, 1, 1, 15, 1, 16, 4, 17, 5, 18, 1, 19, 6, 20, 1, 21, 1, 22, 5, 23, 1, 24, 2, 25, 7, 2, 1, 26, 8, 27, 6, 28, 1, 29, 1, 30, 9, 31, 2, 1, 1, 32, 7, 3, 1, 33, 1, 34, 10, 4, 8, 35, 1, 36, 11, 37, 1, 38, 9, 39, 12, 40, 1, 41, 2, 42, 10, 43, 13

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A049559.

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

Cf. A000010, A049559, A209211.

Cf. also A081373, A303756, A330756.

b[_] = 0;
a[n_] := With[{t = GCD[n-1, EulerPhi[n]]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 27 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A049559(n) = gcd(n-1, eulerphi(n));
v330747 = ordinal_transform(vector(up_to, n, A049559(n)));
A330747(n) = v330747[n];

A303753 Ordinal transform of cototient (A051953).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 5, 1, 6, 2, 1, 3, 7, 1, 8, 2, 1, 3, 9, 1, 1, 1, 2, 2, 10, 1, 11, 3, 1, 1, 1, 1, 12, 1, 1, 2, 13, 1, 14, 3, 1, 4, 15, 1, 2, 2, 1, 1, 16, 1, 2, 2, 2, 3, 17, 1, 18, 3, 1, 4, 1, 1, 19, 2, 1, 2, 20, 1, 21, 1, 1, 1, 2, 1, 22, 2, 2, 1, 23, 1, 3, 2, 1, 3, 24, 1, 2, 4, 1, 5, 1, 1, 25, 1, 1, 2, 26, 1, 27, 2, 1

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Number of values of k, 1 <= k <= n, with A051953(k) = A051953(n).

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

Cf. A051953, A065385 (gives a subset of the positions of ones).

Cf. also A081373, A303754.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= numtheory[phi](n)-n; b(t):= b(t)+1
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 30 2018

b[_] = 0;
a[n_] := a[n] = With[{t = EulerPhi[n]-n}, b[t] = b[t]+1];
Array[a, 120] (* Jean-François Alcover, Dec 19 2021, after Alois P. Heinz *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A051953(n) = (n - eulerphi(n));
v303753 = ordinal_transform(vector(up_to,n,A051953(n)));
A303753(n) = v303753[n];

For all n >= 1, a(A000040(n)) = n.

A303754 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A303753(k) = A303753(n), where A303753 is ordinal transform of cototient, A051953.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 4, 5, 1, 6, 1, 3, 7, 2, 1, 8, 1, 4, 9, 3, 1, 10, 11, 12, 5, 6, 1, 13, 1, 4, 14, 15, 16, 17, 1, 18, 19, 7, 1, 20, 1, 5, 21, 2, 1, 22, 8, 9, 23, 24, 1, 25, 10, 11, 12, 6, 1, 26, 1, 7, 27, 3, 28, 29, 1, 13, 30, 14, 1, 31, 1, 32, 33, 34, 15, 35, 1, 16, 17, 36, 1, 37, 8, 18, 38, 9, 1, 39, 19, 4, 40, 2, 41, 42, 1, 43, 44, 20, 1, 45, 1, 21

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of f, where f(1) = 0 and f(n) = A303753(n) for n > 1.

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

Cf. A051953, A303753.

Cf. also A081373, A303757.

b[_] = 0;
A303753[n_] := A303753[n] = With[{t = EulerPhi[n] - n}, b[t] = b[t]+1];
f[n_] := If[n == 1, 0, A303753[n]];
Clear[b]; b[_] = 0;
a[n_] := a[n] = With[{t = f[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 19 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A051953(n) = (n - eulerphi(n));
v303753 = ordinal_transform(vector(up_to,n,A051953(n)));
Aux303754(n) = if(1==n,0,v303753[n]);
v303754 = ordinal_transform(vector(up_to,n,Aux303754(n)));
A303754(n) = v303754[n];

A303755 Ordinal transform of A289625.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 1, 3, 1, 3, 2, 1, 1, 4, 1, 2, 3, 2, 1, 2, 2, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 2, 2, 2, 1, 6, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 7, 1, 2, 2, 2, 1, 1, 1, 4, 3, 3, 1, 4, 1, 1, 2

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Equally, ordinal transform of A289626.

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

Cf. A289625, A289626.

Cf. also A081373, A303756.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
v303755 = ordinal_transform(vector(up_to,n,A289625(n)));
A303755(n) = v303755[n];

A322874 Ordinal transform of A007431, which is Möbius transform of Euler phi.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 1, 1, 1, 3, 1, 4, 1, 4, 2, 2, 1, 5, 1, 3, 2, 6, 1, 2, 1, 7, 1, 3, 1, 8, 1, 1, 2, 9, 2, 3, 1, 10, 2, 1, 1, 11, 1, 3, 2, 12, 1, 4, 1, 13, 3, 3, 1, 14, 2, 1, 2, 15, 1, 4, 1, 16, 1, 2, 1, 17, 1, 4, 2, 18, 1, 2, 1, 19, 3, 3, 2, 20, 1, 3, 2, 21, 1, 4, 3, 22, 3, 1, 1, 23, 1, 3, 2, 24, 2, 3, 1, 25, 3, 4, 1, 26, 1, 1, 5

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A007431, A081373, A303756, A322873.

A007431[n_] := Sum[EulerPhi[d] MoebiusMu[n/d], {d, Divisors[n]}];
b[_] = 0;
a[n_] := a[n] = With[{t = A007431[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A007431(n) = sumdiv(n,d,moebius(n/d)*eulerphi(d));
v322874 = ordinal_transform(vector(up_to,n,A007431(n)));
A322874(n) = v322874[n];

A330739 Number of values of k, 1 <= k <= n, with A047994(k) = A047994(n), where A047994 is unitary totient function uphi(n).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 3, 1, 2, 3, 2, 1, 1, 1, 4, 1, 2, 1, 4, 1, 1, 1, 2, 2, 3, 1, 3, 4, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 5, 2, 2, 1, 2, 2, 2, 3, 3, 1, 6, 1, 4, 2, 1, 3, 2, 1, 4, 1, 7, 1, 1, 1, 4, 5, 1, 2, 8, 1, 3, 1, 3, 1, 5, 1, 3, 2, 2, 1, 3, 2, 2, 4, 2, 3, 1, 1, 6, 2, 4, 1, 4, 1, 1, 7

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A047994.

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

Cf. A047994.

Cf. also A081373 (ordinal transform of Euler totient function phi), A331177.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v330739 = ordinal_transform(vector(up_to, n, A047994(n)));
A330739(n) = v330739[n];

A331178 Number of values of k, 1 <= k <= n, with A023900(k) = A023900(n), where A023900 is Dirichlet inverse of Euler totient function phi.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 2, 3, 2, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 6, 2, 4, 1, 3, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 1, 2, 1, 1, 1, 8, 1, 2, 3, 2, 1, 2, 1, 5, 4, 2, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 2, 1, 2, 9, 1, 4, 2, 6, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A023900.

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

Cf. A000010, A023900.

Cf. also A081373, A331175, A331179, A331180.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
v331178 = ordinal_transform(vector(up_to, n, A023900(n)));
A331178(n) = v331178[n];

A331182 Number of values of k, 1 <= k <= n, with A083254(k) = A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 3, 4, 1, 1, 1, 2, 3, 4, 1, 1, 2, 5, 2, 3, 1, 1, 1, 5, 1, 6, 1, 1, 1, 7, 3, 2, 1, 1, 1, 4, 4, 8, 1, 1, 2, 1, 2, 5, 1, 2, 1, 3, 3, 9, 1, 1, 1, 10, 4, 6, 1, 1, 1, 6, 1, 1, 1, 1, 1, 11, 2, 7, 1, 1, 1, 2, 2, 12, 1, 1, 2, 13, 2, 4, 1, 1, 1, 8, 3, 14, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A083254.

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

Cf. A000010, A083254.

Cf. also A081373, A331175, A331181.

Module[{b}, b[_] = 0;
a[n_] := With[{t = 2 EulerPhi[n] - n}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A083254(n) = (2*eulerphi(n)-n);
v331182 = ordinal_transform(vector(up_to,n,A083254(n)));
A331182(n) = v331182[n];

A069823 Nonprime numbers k for which there is no x < k such that phi(x) = phi(k).

Original entry on oeis.org

1, 15, 25, 35, 51, 65, 69, 81, 85, 87, 121, 123, 129, 141, 143, 159, 161, 177, 185, 187, 203, 213, 235, 247, 249, 253, 255, 265, 267, 275, 289, 299, 301, 309, 321, 323, 339, 341, 343, 361, 393, 403, 415, 425, 447, 485, 489, 501, 519, 527, 529, 535, 537, 551

Offset: 1

Benoit Cloitre, Apr 28 2002

If p is prime there is no x < p such that phi(x) = phi(p) = p-1 since phi(x) < p-1.
Nonprime numbers k such that A081373(k)=1; i.e., the number of numbers not exceeding k, and with identical phi value to that of k, equals one. - Labos Elemer, Mar 24 2003
For 1 < n, if a(n) is squarefree, then phi(a(n)) is nonsquarefree. The converse is also true: for 1 < n, if phi(a(n)) is squarefree then a(n) is nonsquarefree. - Torlach Rush, Dec 26 2017

			k=25, a nonprime; phi values for k <= 25 are {1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20}; no phi(k) except phi(25) equals 20, A081373(25)=1, so 25 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A081373, A067004, A000010.

f[x_] := EulerPhi[x] fc[x_] := Count[Table[f[j]-f[x], {j, 1, x}], 0] t1=Flatten[Position[Table[fc[w], {w, 1, 1000}], 1]] t2=Flatten[Position[PrimeQ[t1], False]] Part[t1, t2]
(* Second program: *)
Union@ Select[Values[PositionIndex@ Array[EulerPhi, 600]][[All, 1]], ! PrimeQ@ # &] (* Michael De Vlieger, Dec 31 2017 *)

for(s=1,600,if((1-isprime(s))*abs(prod(i=1,s-1,eulerphi(i)-eulerphi(s)))>0, print1(s,",")))

cp's OEIS Frontend

A322873 Ordinal transform of A300721, which is Möbius transform of A060681.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A330747 Number of values of k, 1 <= k <= n, with A049559(k) = A049559(n), where A049559(n) = gcd(n - 1, phi(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A303753 Ordinal transform of cototient (A051953).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A303754 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A303753(k) = A303753(n), where A303753 is ordinal transform of cototient, A051953.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A303755 Ordinal transform of A289625.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A322874 Ordinal transform of A007431, which is Möbius transform of Euler phi.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A330739 Number of values of k, 1 <= k <= n, with A047994(k) = A047994(n), where A047994 is unitary totient function uphi(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A331178 Number of values of k, 1 <= k <= n, with A023900(k) = A023900(n), where A023900 is Dirichlet inverse of Euler totient function phi.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A331182 Number of values of k, 1 <= k <= n, with A083254(k) = A083254(n), where A083254(n) = 2*phi(n) - n.

Views