A081373 -id:A081373 - cp's OEIS frontend

A272328 Number of integers 1<=k<=n such that phi(n)=phi(n+k) where phi is Euler's totient function A000010.

1, 0, 2, 1, 2, 0, 2, 2, 2, 1, 1, 0, 2, 1, 4, 3, 2, 0, 2, 2, 4, 0, 1, 1, 3, 3, 2, 2, 1, 0, 1, 4, 3, 3, 5, 1, 3, 1, 6, 2, 3, 0, 2, 2, 7, 0, 1, 1, 2, 1, 5, 6, 1, 0, 5, 5, 5, 0, 1, 0, 4, 0, 5, 5, 4, 0, 1, 4, 2, 4, 1, 3, 6, 4, 6, 3, 5, 2, 1, 3, 1, 5, 1, 1, 4, 1, 2

Offset: 1

Tom Edgar, Apr 25 2016

nonn

If n is odd, then phi(n) = phi(2n) so that a(n)>=1.
If n is a member of A043343, then a(n)=0.
It seems that every nonnegative integer appears in this sequence.

			For n=2: phi(2) = 1; whereas phi(2+1) = 2 and phi(2+2) = 2. Thus a(2) = 0.
For n=5: phi(5) = 4, phi(5+1)=2, phi(5+2)=6, phi(5+3) = 4, phi(5+4) = 6, and phi(5+5) = 4. Since phi(5) = phi(5+3) = phi(5+5), a(5) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A043343, A066659, A081373.

Table[Count[Range@ n, k_ /; EulerPhi@ n == EulerPhi[n + k]], {n, 120}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = my(x=eulerphi(n)); sum(k=1, n, eulerphi(n+k) == x); \\ Michel Marcus, Mar 08 2020

from sympy import totient
nmax = 10**4
philist = [totient(i) for i in range(1,2*nmax+1)]
A272328_list = [philist[i+1:2*(i+1)].count(philist[i]) for i in range(nmax)] # Chai Wah Wu, Apr 26 2016

[sum([1 for k in [1..n] if euler_phi(n)==euler_phi(n+k)]) for n in [1..1000]]

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];

A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A063994.

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

Cf. A063994, A209211.

Cf. also A081373, A303756, A330747.

A063994[n_] := If[n==1, 1, Times @@ GCD[n-1, First /@ FactorInteger[n]-1]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A063994[n]}, b[t] = b[t]+1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
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; };
v330756 = ordinal_transform(vector(up_to, n, A063994(n)));
A330756(n) = v330756[n];

A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A173557.

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

Cf. A173557.

Cf. also A081373, A331175, A331178.

A173557[n_] := If[n == 1, 1, Times @@ (FactorInteger[n][[All, 1]] - 1)];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A173557[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; };
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
v331179 = ordinal_transform(vector(up_to, n, A173557(n)));
A331179(n) = v331179[n];

A081376 a(n) is the least number such that A067003[a(n)] = n.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 122, 123

Offset: 1

Labos Elemer, Mar 24 2003

nonn

Cf. A067003, A081373-A081375, A001221.

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

A171934 Backwards van Eck transform of A000010.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 3, 2, 2, 0, 2, 0, 5, 0, 1, 0, 4, 0, 4, 8, 11, 0, 4, 0, 5, 8, 2, 0, 6, 0, 15, 8, 2, 0, 8, 0, 11, 4, 6, 0, 6, 0, 11, 6, 23, 0, 8, 6, 6, 0, 7, 0, 16, 14, 4, 20, 29, 0, 12, 0, 31, 6, 13, 0, 16, 0, 4, 0, 14, 0, 2, 0, 11, 20, 2, 16, 6, 0, 12, 0, 7, 0, 6, 0, 37, 0, 6, 0, 6, 18, 23, 16, 47

Offset: 1

N. J. A. Sloane, Oct 24 2010

nonn

Given a sequence a, the backwards van Eck transform b is defined as follows: If a(n) has already appeared in a, let a(m) be the most recent occurrence, and set b(n)=n-m; otherwise b(n)=0. (Comment from A171899).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
N. J. A. Sloane, Transforms

Cf. A000010.

Cf. also A081373, A171899, A171900, A171932, A171933, A171936, A171941.

Block[{a = Array[EulerPhi, 94], b = {}, m}, Do[If[! IntegerQ[m[#]], Set[m[#], i]; AppendTo[b, 0], AppendTo[b, i - m[#]]; Set[m[#], i]] &@ a[[i]], {i, Length[a]}]; b] (* Michael De Vlieger, Apr 06 2021 *)

up_to = 105;
backVanEck_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] = i-pp, outvec[i] = 0); mapput(om,invec[i],i)); outvec; };
v171934 = backVanEck_transform(vector(up_to,n,eulerphi(n)));
A171934(n) = v171934[n]; \\ Antti Karttunen, Apr 06 2021

A344772 Ordinal transform of infinitary phi, A091732.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2021

Number of values of k, 1 <= k <= n, with A091732(k) = A091732(n).

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

Cf. A091732, A050376, A302777.

Cf. also A081373 (ordinal transform of phi), A303756 (of Carmichael's lambda), A330739 (of unitary phi).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1);
b[_] = 0;
a[n_] := a[n] = With[{t = iphi[n]}, b[t] = b[t] + 1];
Array[a, 105] (* Jean-François Alcover, Dec 27 2021, after Amiram Eldar in A091732 *)

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; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
v344772 = ordinal_transform(vector(up_to,n,A091732(n)));
A344772(n) = v344772[n];

cp's OEIS Frontend

A272328 Number of integers 1<=k<=n such that phi(n)=phi(n+k) where phi is Euler's totient function A000010.

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A322025 Ordinal transform of A322023.

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

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A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

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A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

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A081376 a(n) is the least number such that A067003[a(n)] = n.

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A171934 Backwards van Eck transform of A000010.

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A344772 Ordinal transform of infinitary phi, A091732.

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