This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a := function(n)
local S, T, R, max, i, k, r;
S:=[];
for i in DivisorsInt(n)+1 do
if IsPrime(i)=true then
S:=Concatenation(S,[i]);
fi;
od;
T:=[];
for k in [1..Size(S)] do
T:=Concatenation(T,[S[k]/(S[k]-1)]);
od;
max := n*Product(T);
R:=[];
for r in [1..Int(max)] do
if Phi(r)=n then
R:=Concatenation(R,[r]);
fi;
od;
return Size(R);
end; # Miles Englezou, Oct 22 2024
[#EulerPhiInverse(n): n in [1..100]]; // Marius A. Burtea, Sep 08 2019
with(numtheory): A014197:=n-> nops(invphi(n)): seq(A014197(n), n=1..200);
a[1] = 2; a[m_?OddQ] = 0; a[m_] := Module[{p, nmax, n, k}, p = Select[ Divisors[m]+1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; k = 0; While[n <= nmax, If[EulerPhi[n] == m, k++]; n++]; k]; Array[a, 92] (* Jean-François Alcover, Dec 09 2011, updated Apr 25 2016 *)
With[{nn = 116}, Function[s, Function[t, Take[#, nn] &@ ReplacePart[t, Map[# -> Length@ Lookup[s, #] &, Keys@ s]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(3/2)] + 10]] (* Michael De Vlieger, Jul 19 2017 *)
A014197(n,m=1) = { n==1 && return(1+(m<2)); my(p,q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0,valuation(q=n\d,p=d+1), A014197(q\p^i,p))))} \\ M. F. Hasler, Oct 05 2009
a(n) = invphiNum(n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp
from sympy import totient, divisors, isprime, prod
def a(m):
if m == 1: return 2
if m % 2: return 0
X = (x + 1 for x in divisors(m))
nmax=m*prod(i/(i - 1) for i in X if isprime(i))
n=m
k=0
while n<=nmax:
if totient(n)==m:k+=1
n+=1
return k
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 18 2017, after Mathematica code
The six polynomials of degree 2 consist of 3 irreducible cyclotomic polynomials: x^2+1, x^2+x+1 and x^2-x+1 and 3 products of 2 linear cyclotomic polynomials: x^2+2x+1, x^2-1 and x^2-2x+1. The six plane crystallographic operations are the identity operation, rotations by 2 Pi/k with k = 2,3,4,6, and a reflection.
with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jul 04 2019
terms = 40;
S[m_] := S[m] = CoefficientList[Product[1/(1 - x^EulerPhi[k]),
{k, 1, m*terms}] + O[x]^terms, x];
S[m = 1];
S[m++];
While[S[m] != S[m-1], m++];
S[m] (* Jean-François Alcover, Apr 14 2017, after Christopher J. Smyth, updated May 13 2022 *)
a(3) = 6 because there are six degree-3 products of distinct cyclotomic polynomials, namely (z-1)(z^2+z+1), (z-1)(z^2+1), (z-1)(z^2-z+1), (z+1)(z^2+z+1), (z+1)(z^2+1) and (z+1)(z^2-z+1).
Table[SeriesCoefficient[Product[(1 + x^EulerPhi@ i), {i, n E^2}], {x, 0, n}], {n, 0, 92}] (* Michael De Vlieger, Jan 10 2017 *)
with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(
g(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
end:
a:= n-> g(n)-g(n-1):
seq(a(n), n=0..40); # Alois P. Heinz, Jun 23 2023
nt = 100; (* number of terms *)
f[kmax_] := f[kmax] = CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 2, kmax}] + O[x]^nt, x]; f[kmax = nt]; f[kmax += nt];
While[f[kmax] != f[kmax - nt], kmax += nt];
f[kmax] (* Jean-François Alcover, Nov 29 2023 *)
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