A303756 -id:A303756 - cp's OEIS frontend

A322023 Lexicographically earliest such sequence a that a(i) = a(j) => A081373(i) = A081373(j) and A303756(i) = A303756(j), for all i, j. Here A081373 and A303756 are the ordinal transforms of Euler phi and Carmichael lambda.

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

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of the ordered pair [A081373(n), A303756(n)].

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

Cf. A000010, A002322, A081373, A303756, A319339, A322024, A322025 (ordinal transform).

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; };
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; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
v303756 = ordinal_transform(vector(up_to,n,A002322(n)));
A303756(n) = v303756[n];
v322023 = rgs_transform(vector(up_to, n, [A081373(n), A303756(n)]));
A322023(n) = v322023[n];

A081373 Number of values of k, 1 <= k <= n, with phi(k) = phi(n), where phi is Euler totient function, A000010.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 1, 2, 1, 4, 1, 3, 2, 2, 1, 4, 1, 3, 2, 4, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 4, 1, 4, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 3, 2, 1, 5, 1, 3, 1, 6, 1, 7, 1, 4, 3, 5, 2, 8, 1, 4, 1, 4, 1, 9, 1, 3, 1, 5, 1, 10, 2, 2, 3, 2, 3, 5, 1, 4, 4, 6

Offset: 1

Labos Elemer, Mar 24 2003

nonn

Ordinal transform of Euler totient function phi, A000010. - Antti Karttunen, Aug 26 2024

			For n = 16: phi(k) = {1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8} for k = 1,...,n; 2 numbers exist with phi(k) = phi(n) = 8: {15,16}, so a(16) = 2.
If n = p is an odd prime number, then a(p) = 1 with phi(k) = p-1.

Paul Tek, Table of n, a(n) for n = 1..100000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A081375 (positions of records), A210719 (of 1's).

Cf. also A067004, A303756, A303757, A303777 (ordinal transform of this sequence).

f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] Table[f[w], {w, 1, 100}]

a(n)=my(t=eulerphi(n), s); sum(k=1, n, eulerphi(k)==t) \\ Charles R Greathouse IV, Feb 21 2013, corrected by Antti Karttunen, Aug 26 2024

a(n) = #select(x -> x <= n, invphi(eulerphi(n))); \\ Amiram Eldar, Nov 08 2024, using Max Alekseyev's invphi.gp

A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A032742, A060681, A081373, A303756, A322870, A322872, A322873, A322874.

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
b[_] = 1;
a[n_] := a[n] = With[{t = A060681[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]))));
v322871 = ordinal_transform(vector(up_to,n,A060681(n)));
A322871(n) = v322871[n];

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

Original entry on oeis.org

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];

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];

A303758 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of f, where f(1) = 0 and f(n) = A002322(n) for n > 1.

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

Cf. A002322.

Cf. also A303756, A303757.

a[1] = 1; a[n_] := With[{c = CarmichaelLambda[n]}, Select[Range[2, n], c == CarmichaelLambda[#]&] // Length];
Array[a, 1000] (* Jean-François Alcover, Sep 19 2020 *)

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; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
Aux303758(n) = if(1==n,0,A002322(n));
v303758 = ordinal_transform(vector(up_to,n,Aux303758(n)));
A303758(n) = v303758[n];

Except for a(2) = 1, a(n) = A303756(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];

A322025 Ordinal transform of A322023.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 4, 1, 3, 1, 5, 1, 6, 2, 1, 2, 7, 1, 8, 1, 1, 4, 9, 1, 10, 2, 5, 2, 11, 1, 12, 1, 1, 3, 2, 1, 13, 3, 3, 1, 14, 1, 15, 1, 2, 6, 16, 1, 7, 1, 3, 3, 17, 2, 2, 1, 2, 8, 18, 1, 19, 9, 1, 4, 1, 1, 20, 3, 4, 2, 21, 1, 22, 2, 2, 2, 3, 1, 23, 1, 24, 3, 25, 1, 1, 4, 5, 3, 26, 1, 1, 5, 3, 10, 5, 1, 27, 3, 2, 1, 28, 1, 29, 1, 1

Offset: 1

Antti Karttunen, Dec 01 2018

nonn

Positions where 1, 2, 3, 4, 5, ... occur for the first time are 1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, ... Note that this is not a subsequence of A000961; for example, 187 = 11*17 is a semiprime.

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

Cf. A000010, A002322, A081373, A303753, A303755, A303756, A303777, A322023.

\\ Needs also code from A322023.
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; };
v322025 = ordinal_transform(v322023);
A322025(n) = v322025[n];

A322872 Ordinal transform of A171462, where A171462(n) = n - A052126(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A052126, A171462, A081373, A303756, A322871.

A171462[n_] := If[n == 1, 0, Module[{f = FactorInteger[n], p},
     p = f[[-1, 1]]; n(p-1)/p]];
b[_] = 0;
a[n_] := a[n] = With[{t = A171462[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 21 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; };
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
v322872 = ordinal_transform(vector(up_to,n,A171462(n)));
A322872(n) = v322872[n];

cp's OEIS Frontend

A322023 Lexicographically earliest such sequence a that a(i) = a(j) => A081373(i) = A081373(j) and A303756(i) = A303756(j), for all i, j. Here A081373 and A303756 are the ordinal transforms of Euler phi and Carmichael lambda.

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A081373 Number of values of k, 1 <= k <= n, with phi(k) = phi(n), where phi is Euler totient function, A000010.

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A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

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

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

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A303755 Ordinal transform of A289625.

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A303758 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

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Formula

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

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A322025 Ordinal transform of A322023.