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.
Triangle: 0, 1, 2, 3, 4, 6, 5, 8, 10, 12, 15, 16, 20, 24, 30, 13, 21, 26, 28, 36, 42, 51, 64, 68, 80, 96, 102, 120, 37, 57, 63, 74, 76, 108, 114, 126, 41, 55, 75, 82, 88, 100, 110, 132, 150
Needs["CNT`"]; nn = 10; t = Table[{}, {nn}]; n = 0; t[[1]] = {0}; left = nn - 1; While[left > 0, n++; p = PhiInverse[n]; cnt = Length[p]; If[cnt <= nn && t[[cnt]] == {}, t[[cnt]] = p; left--]]; t
Needs["CNT`"]; nn = 50; t = Table[0, {nn}]; n = 0; left = nn - 1; While[left > 0, n++; cnt = Length[PhiInverse[n]]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; left--]]; Join[{0}, Table[PhiInverse[n][[-1]], {n, Rest[t]}]]
a(4) = 8 since phi(x) = 8 has the solutions {15, 16, 20, 24, 30}, one more solution than a(3) = 4 for which phi(x) = 4 has solutions {5, 8, 10, 12}.
HighlyTotientNumbers := proc(n) # n > 1 is search maximum local L, m, i, r; L := NULL; m := 0; for i from 1 to n do r := nops(numtheory[invphi](i)); if r > m then L := L,[i,r]; m := r fi od; [L] end: A097942_list := n -> seq(s[1], s = HighlyTotientNumbers(n)); A097942_list(500); # Peter Luschny, Sep 01 2012
searchMax = 2000; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; highlyTotientList = {1}; currHigh = 1; Do[If[phiAnsYldList[[n]] > phiAnsYldList[[currHigh]], highlyTotientList = {highlyTotientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyTotientList]
{ A097942_list(n) = local(L, m, i, r);
m = 0;
for(i=1, n,
\\ from Max Alekseyev, http://home.gwu.edu/~maxal/gpscripts/
r = numinvphi(i);
if(r > m, print1(i,", "); m = r) );
} \\ Peter Luschny, Sep 01 2012
def HighlyTotientNumbers(n) : # n > 1 is search maximum.
R = {}
for i in (1..n^2) :
r = euler_phi(i)
if r <= n :
R[r] = R[r] + 1 if r in R else 1
# print R.keys() # A002202
# print R.values() # A058277
P = []; m = 1
for l in sorted(R.keys()) :
if R[l] > m : m = R[l]; P.append((l,m))
# print [l[0] for l in P] # A097942
# print [l[1] for l in P] # A131934
return P
A097942_list = lambda n: [s[0] for s in HighlyTotientNumbers(n)]
A097942_list(500) # Peter Luschny, Sep 01 2012
a(3) = 8 since x - phi(x) = 8 has three solutions, {12, 14, 16}, one more than a(2) = 4 which has two solutions, {6, 8}.
searchMax = 4000; coPhiAnsYldList = Table[0, {searchMax}]; Do[coPhiAns = m - EulerPhi[m]; If[coPhiAns <= searchMax, coPhiAnsYldList[[coPhiAns]]++ ], {m, 1, searchMax^2}]; highlyCototientList = {2}; currHigh = 2; Do[If[coPhiAnsYldList[[n]] > coPhiAnsYldList[[currHigh]], highlyCototientList = {highlyCototientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyCototientList]
a(n) = if (n==1, 0, my(k=1); while (#invphi(k) != n, k++); k); \\ using invphi in PARI scripts link; Michel Marcus, Oct 09 2023
solnum[n_] := Length[invUPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k] + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invUPhi from A361966 *)
83 is a term because the three solutions (249,332,498) to phi(x) = 164 can be written as (3*83, 4*83, 6*83).
import Data.List.Ordered (insertBag)
import Data.List (groupBy); import Data.Function (on)
a085713 n = a085713_list !! (n-1)
a085713_list = 1 : r yx3ss where
r (ps:pss) | a010051' cd == 1 &&
map (flip div cd) ps == [3, 4, 6] = cd : r pss
| otherwise = r pss where cd = foldl1 gcd ps
yx3ss = filter ((== 3) . length) $
map (map snd) $ groupBy ((==) `on` fst) $
f [1..] a002110_list []
where f is'@(i:is) ps'@(p:ps) yxs
| i < p = f is ps' $ insertBag (a000010' i, i) yxs
| otherwise = yxs' ++ f is' ps yxs''
where (yxs', yxs'') = span ((<= a000010' i) . fst) yxs
-- Reinhard Zumkeller, Nov 25 2015
t = Table[ EulerPhi[n], {n, 1, 5000}]; u = Union[ Select[t, Count[t, # ] == 3 &]]; a = {}; Do[k = 1; While[ EulerPhi[3k] != u[[n]], k++ ]; AppendTo[a, k], {n, 1, 60}]; Sort[a]
is(p) = if(p > 1 && !isprime(p), 0, invphi(eulerphi(3*p)) == [3*p, 4*p, 6*p]); \\ Amiram Eldar, Nov 19 2024, using Max Alekseyev's invphi.gp
a(3) = 4 because there are 3 even solutions (8, 10, 12) of phi(x) = 4 and for all k < 4 the number of even solutions of phi(x) = k is unequal to 3.
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